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<title>L-Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">L-Funktion</h1>

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L-Funktionen werden in der <a href="Analytische_Zahlentheorie" title="Analytische Zahlentheorie">analytischen Zahlentheorie</a> und darauf aufbauenden mathematischen Gebieten untersucht. Das prototypische Beispiel einer L-Funktion ist die <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannsche Zeta-Funktion</a>. L-Funktionen haben fundamentale Eigenschaften mit der Riemannschen Zeta-Funktion gemeinsam. Sie sind also Verallgemeinerungen der Riemannschen Zeta-Funktion. Zu den fundamentalen Eigenschaften der Riemannschen Zeta-Funktion zählen:

<ul><li>die Riemannsche Zeta-Funktion stimmt in einem Teilbereich der <a href="Komplexe_Zahlenebene" class="mw-redirect" title="Komplexe Zahlenebene">komplexen Zahlenebene</a> mit einer <a href="Dirichlet-Reihe" class="mw-redirect" title="Dirichlet-Reihe">Dirichlet-Reihe</a> und einem <a href="Eulersches_Produkt" class="mw-redirect" title="Eulersches Produkt">Euler-Produkt</a> überein, die beide <a href="Absolute_Konvergenz" class="mw-redirect" title="Absolute Konvergenz">absolut konvergieren</a>;</li>

<li>die zunächst nur in jenem Teilbereich definierte Riemannsche Zeta-Funktion lässt sich <a href="Analytische_Fortsetzung" title="Analytische Fortsetzung">analytisch fortsetzen</a> zu einer auf der komplexen Zahlenebene <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphen Funktion</a>;</li>

<li>die fortgesetzte Riemannsche Zeta-Funktion genügt einer <a href="Funktionalgleichung" title="Funktionalgleichung">Funktionalgleichung</a> eines bestimmten Typs.</li></ul>

Basierend auf den grundlegenden Arbeiten von <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (1707–1783) zur heute so bezeichneten Riemannschen Zeta-Funktion, untersuchten die Mathematiker <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (1826–1866), <a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Dirichlet</a> (1805–1859), <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> (1831–1916), <a href="Erich_Hecke" title="Erich Hecke">Erich Hecke</a> (1887–1947) und <a href="Emil_Artin" title="Emil Artin">Emil Artin</a> (1898–1962) grundlegende Unterklassen von L-Funktionen, die heute deren jeweiligen Namen tragen.

Die forschende Suche nach einer allgemeinen und eindeutigen Definition des Begriffs „L-Funktion“, welche die gewünschten und zum Teil noch unbewiesenen Eigenschaften von L-Funktionen beweisbar macht, ist noch nicht abgeschlossen. Vielmehr handelt es sich um ein wichtiges Ziel der analytischen Zahlentheorie, Klarheit über die sinnvollste Definition des Begriffs „L-Funktion“ zu gewinnen. In dieser Richtung hat <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> (1917–2007) im Jahr 1989 eine axiomatische Definition der Klasse aller L-Funktionen vorgeschlagen, die heute den Namen „<a href="Selberg-Klasse" title="Selberg-Klasse">Selberg-Klasse</a>“ trägt.<a href="#cite_note-1">[1]</a> Ob diese oder andere Definitionsvorschläge schon alle wünschenswerten Eigenschaften von L-Funktionen umfassen und unerwünschte ausschließen, ist noch nicht abschließend geklärt. Nach wie vor prägen mathematische Vermutungen (d. h. unbewiesene, aber für plausibel oder zumindest wünschenswert gehaltene Aussagen über Eigenschaften von L-Funktionen) die Theorie der L-Funktionen. Diese zählt somit weiterhin zu den Gebieten intensiver mathematischer Forschung.

Die beiden Begriffe „L-Funktion“ und „Zeta-Funktion“ werden häufig synonym verwendet. Trotzdem zählen nicht alle mathematischen Funktionen, deren Namen den Begriff „Zeta-Funktion“ enthalten, zu den L-Funktionen. Beispielsweise gehört die <a href="Primzetafunktion" title="Primzetafunktion">Primzetafunktion</a> nicht zu den L-Funktionen, da sie analytisch nicht auf die ganze komplexe Ebene fortgesetzt werden kann.

Ein erstes Verständnis des Themenbereichs der L-Funktionen erfordert mathematische Kenntnisse im Bereich der <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a>, der <a href="Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a>, der <a href="Analytische_Zahlentheorie" title="Analytische Zahlentheorie">analytischen</a> und <a href="Algebraische_Zahlentheorie" title="Algebraische Zahlentheorie">algebraischen Zahlentheorie</a> sowie der <a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a> von <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a>. Solche Vorkenntnisse können in diesem Artikel zwar teilweise erläutert, aber nicht umfassend dargestellt werden.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Wie in der Einleitung erwähnt, gibt es noch keine allgemeine, eindeutige und weithin anerkannte Definition des Begriffs „L-Funktion“. Der nachfolgende Definitionsansatz folgt dem Ansatz, den die beiden Mathematiker <a href="Henryk_Iwaniec" title="Henryk Iwaniec">Henryk Iwaniec</a> und <a href="Emmanuel_Kowalski" title="Emmanuel Kowalski">Emmanuel Kowalski</a> in ihrem Lehrbuch zur analytischen Zahlentheorie angegeben haben.<a href="#cite_note-2">[2]</a>

Dieser Definitionsansatz ist zwar stellenweise abstrakt und unvollständig in dem Sinne, dass er die „arithmetischen Objekte“, denen er eine „L-Funktion“ zuordnet, sowie den genauen Mechanismus dieser Zuordnung nicht näher spezifiziert. Er umfasst aber die Eigenschaften, die von L-Funktionen im Allgemeinen erwartet werden, und ermöglicht es somit, die entscheidenden Merkmale dieser Funktionen zu erläutern. Nebenbei werden auch noch weitere Grundbegriffe der Theorie der L-Funktionen eingeführt:

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> ein – im Rahmen dieser abstrakten Definition nicht näher spezifiziertes – arithmetisches Objekt, z. B. ein <a href="Dirichlet-Charakter" class="mw-redirect" title="Dirichlet-Charakter">Dirichlet-Charakter</a> oder ein <a href="Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">algebraischer Zahlkörper</a>. Diesem arithmetischen Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet ist eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy">, die komplexe Argumente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle s\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f030d5fc9630ec99be496c46bc45c53c34033e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle \textstyle s\in \mathbb {C} }" loading="lazy"> auf komplexe Funktionswerte abbildet. Iwaniec und Kowalski nennen eine solche Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> eine L-Funktion, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> die nachfolgenden, mathematischen Objekte zugeordnet sind (siehe D-1 bis D-6), die die anschließend genannten Bedingungen erfüllen (siehe B-1 bis B-9):

D-1: Dirichlet-Reihe und Euler-Produkt

Dem arithmetischen Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet sind eine <a href="Dirichlet-Reihe" class="mw-redirect" title="Dirichlet-Reihe">Dirichlet-Reihe</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {N} }\lambda (f,n)n^{-s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeea792778b107a54e1ad74a78815549ac1e348e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.29ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in \mathbb {N} }\lambda (f,n)n^{-s}}" loading="lazy">,</dd></dl>

welche man auch eine L-Reihe nennt, und ein <a href="Eulersches_Produkt" class="mw-redirect" title="Eulersches Produkt">Euler-Produkt</a>

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<annotation encoding="application/x-tex">{\displaystyle \prod _{p\in \mathbb {P} }(1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a69d1acb1b8280b30ab043a31b908d48ff8d3863.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.042ex; width:47.99ex; height:5.843ex;" alt="{\displaystyle \prod _{p\in \mathbb {P} }(1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}}" loading="lazy">.</dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (f,n)\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (f,n)\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66759d817ed64dcadf29958dbb9e3e9b86f27693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.39ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (f,n)\in \mathbb {C} }" loading="lazy"> für alle <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3579ec499388d52fefca601a35c16ab65ee6e4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle \textstyle n\in \mathbb {N} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (f,1)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (f,1)=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3b7eb7049b4cc8349f155352cda370a6975c49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.9ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (f,1)=1}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {P} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {P} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52b0f93b18ee43348773fe476332c1dc86ebf3ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \textstyle \mathbb {P} }" loading="lazy"> symbolisiert die Menge aller <a href="Primzahl" title="Primzahl">Primzahlen</a>. Die natürliche Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle d\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle d\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8681a9af02c072b394c977c7a7a7d86ff3bd3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.735ex; height:2.176ex;" alt="{\displaystyle \textstyle d\in \mathbb {N} }" loading="lazy"> heißt der Grad des Euler-Produkts oder auch der Grad der L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy">. Für jede Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy"> und jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle i\in \{1,\ldots ,d\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179c563417daf47cb1a5286642c848f00dfaa65d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:2.843ex;" alt="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \alpha _{i}(f,p)\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \alpha _{i}(f,p)\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5db5d117ce3ec929cea2e9b2b63a064a23179e7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.097ex; height:2.843ex;" alt="{\displaystyle \textstyle \alpha _{i}(f,p)\in \mathbb {C} }" loading="lazy">. Die komplexen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \alpha _{i}(f,p)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \alpha _{i}(f,p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b038f808bc9b2283634b50e5b32696596e2663c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.579ex; height:2.843ex;" alt="{\displaystyle \textstyle \alpha _{i}(f,p)}" loading="lazy"> werden Lokale Wurzeln oder auch Lokale Parameter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy"> genannt. Für ein gegebenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

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<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy"> heißt der Ausdruck

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<annotation encoding="application/x-tex">{\displaystyle (1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ad080a5aaa5fe3344a7491ba415a8731122cda9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.978ex; height:3.176ex;" alt="{\displaystyle (1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}}" loading="lazy">,</dd></dl>

also der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-te Faktor im Euler-Produkt, der Euler-Faktor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(f,s)}">

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<annotation encoding="application/x-tex">{\displaystyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2da541849710de48ddd7bc721630d06b530aca71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle L(f,s)}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">.

D-2: Gamma-Faktor

Daneben ist dem Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> ein so genannter Gamma-Faktor

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (f,s)=\pi ^{-ds/2}\prod _{j=1}^{d}\Gamma \left({\frac {s+\kappa _{j}}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55a67cd5d57e31db259deec3d4452ad9b2834112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.713ex; height:7.676ex;" alt="{\displaystyle \gamma (f,s)=\pi ^{-ds/2}\prod _{j=1}^{d}\Gamma \left({\frac {s+\kappa _{j}}{2}}\right)}" loading="lazy"></dd></dl>

zugeordnet, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0e2dd291dbdc6be5ecbf5d4500c5511f2ee644d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \textstyle \Gamma }" loading="lazy"> die <a href="Gamma-Funktion" class="mw-redirect" title="Gamma-Funktion">Gamma-Funktion</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f3ac3a54990c3e941060e44a8850d1c5824f537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \textstyle \pi }" loading="lazy"> die <a href="Kreiszahl" title="Kreiszahl">Kreiszahl</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4a5f1130824df33e2aff957314f14b0f8745ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle \textstyle d}" loading="lazy"> den oben genannten Grad der L-Funktion bezeichnen. Die Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa _{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c048668d5f94a9a03602808368e765b00738c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.249ex; height:2.343ex;" alt="{\displaystyle \textstyle \kappa _{j}}" loading="lazy"> sind komplexe Zahlen. Sie heißen die Lokalen Parameter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> im Unendlichen oder an der unendlichen Primstelle.

D-3: Führer (Konduktor)

Ebenfalls zugeordnet ist dem Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> eine natürliche Zahl

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(f)\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4955d0dc7778de6765edbcb0cac8fb4224ccf01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.676ex; height:2.843ex;" alt="{\displaystyle q(f)\in \mathbb {N} }" loading="lazy">,</dd></dl>

der so genannte Führer oder Konduktor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy">. Primzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy">, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle q(f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle q(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c84999a1ccf3617e49757b1bd51982bb3af2ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.157ex; height:2.843ex;" alt="{\displaystyle \textstyle q(f)}" loading="lazy"> nicht teilen, heißen unverzweigt bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy">.

D-4: Vollständige L-Funktion

Mit Hilfe der Dirichlet-Reihe, des Gamma-Faktors und des Führers, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet sind, definiert man jetzt die so genannte vollständige L-Funktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy">:

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (f,s)=q(f)^{s/2}\gamma (f,s)L(f,s).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2048ce5129663a541465d247c86f25aee7c3b5e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.645ex; height:3.343ex;" alt="{\displaystyle \Lambda (f,s)=q(f)^{s/2}\gamma (f,s)L(f,s).}" loading="lazy"></dd></dl>

D-5: Wurzelzahl

Des Weiteren ist dem Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> eine komplexe Zahl

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon (f)\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/282575fa10ee079f94e6df877883ce3f3f0ce5b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.551ex; height:2.843ex;" alt="{\displaystyle \epsilon (f)\in \mathbb {C} }" loading="lazy"></dd></dl>

zugeordnet. Diese komplexe Zahl heißt die Wurzelzahl von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy">.

D-6: Duales, arithmetisches Objekt

Schließlich ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> noch ein weiteres, arithmetisches Objekt zugeordnet, das im Rahmen dieser abstrakten Definition nicht näher spezifiziert wird. Es wird das Dual von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> genannt und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> bezeichnet. Wie im Fall von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> sind auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> eine Dirichlet-Reihe

</mrow>

</munder>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {N} }\lambda ({\bar {f}},n)n^{-s}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8e4f066b83d0a1573671ce11432bbc34548a71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.711ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in \mathbb {N} }\lambda ({\bar {f}},n)n^{-s}}" loading="lazy">,</dd></dl>

ein Euler-Produkt

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{p\in \mathbb {P} }(1-\alpha _{1}({\bar {f}},p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{\bar {d}}({\bar {f}},p)p^{-s})^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/435cb73085be0fe3e36391f7e337bb824ca5919f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.042ex; width:49.203ex; height:5.843ex;" alt="{\displaystyle \prod _{p\in \mathbb {P} }(1-\alpha _{1}({\bar {f}},p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{\bar {d}}({\bar {f}},p)p^{-s})^{-1}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {d}}\in \mathbb {N} }">

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {d}}\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b49f89028b0b6e14ebbe9fa34ddb72308b3df58c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.1ex; height:2.509ex;" alt="{\displaystyle \textstyle {\bar {d}}\in \mathbb {N} }" loading="lazy">, ein Gamma-Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \gamma ({\bar {f}},s)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \gamma ({\bar {f}},s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa7c138ba70b094e4d2e226e25d6c96bc9dfe080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.895ex; height:3.176ex;" alt="{\displaystyle \textstyle \gamma ({\bar {f}},s)}" loading="lazy"> und ein Führer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle q({\bar {f}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle q({\bar {f}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db79260c1bd2810c8876effc2d16b7cd63988879.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.578ex; height:3.176ex;" alt="{\displaystyle \textstyle q({\bar {f}})}" loading="lazy"> sowie eine vollständige L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Lambda ({\bar {f}},s)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Lambda ({\bar {f}},s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b36da464149abd39549fbb43203bc408f7ca680f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.246ex; height:3.176ex;" alt="{\displaystyle \textstyle \Lambda ({\bar {f}},s)}" loading="lazy"> zugeordnet. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f={\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f={\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfc35c443c2530335ea60ba56d3b5696f320db09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.076ex; height:3.009ex;" alt="{\displaystyle \textstyle f={\bar {f}}}" loading="lazy">, so nennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> selbstdual, was nichts anderes bedeutet als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (f,n)\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (f,n)\in \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7147f05296d9895870a67f2547d53103b7a4ffe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.39ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (f,n)\in \mathbb {R} }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3579ec499388d52fefca601a35c16ab65ee6e4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle \textstyle n\in \mathbb {N} }" loading="lazy">.<a href="#cite_note-3">[3]</a>

Die oben genannten, dem arithmetischen Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordneten Objekte müssen nun die folgenden Bedingungen erfüllen, damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> die Definition einer L-Funktion nach Iwaniec und Kowalski erfüllt:

B-1: Absolutbetrag von lokalen Parametern bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy">

Für jede Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy"> und jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle i\in \{1,\ldots ,d\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179c563417daf47cb1a5286642c848f00dfaa65d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:2.843ex;" alt="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle |\alpha _{i}(f,p)|<p}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle |\alpha _{i}(f,p)|<p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88ca20d1405b09f5a6c680b5266cc6163630d461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.14ex; height:2.843ex;" alt="{\displaystyle \textstyle |\alpha _{i}(f,p)|<p}" loading="lazy">.

B-2: Werte von lokalen Parametern bei unverzweigtem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy">

Für alle Primzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy">, die bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a2e993679be2df35218311ec561f8236a0aca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="{\displaystyle \textstyle L(f,s)}" loading="lazy"> unverzweigt sind, und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle i\in \{1,\ldots ,d\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/179c563417daf47cb1a5286642c848f00dfaa65d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:2.843ex;" alt="{\displaystyle \textstyle i\in \{1,\ldots ,d\}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \alpha _{i}(f,p)\neq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \alpha _{i}(f,p)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2db57ee03a1ca831c4a7251b723f855d08414f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.839ex; height:2.843ex;" alt="{\displaystyle \textstyle \alpha _{i}(f,p)\neq 0}" loading="lazy">.

B-3: Anforderungen an die lokalen Parameter im Unendlichen

Die Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa _{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa _{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c048668d5f94a9a03602808368e765b00738c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.249ex; height:2.343ex;" alt="{\displaystyle \textstyle \kappa _{j}}" loading="lazy"> sind entweder <a href="Reelle_Zahl" title="Reelle Zahl">reell</a> oder kommen in Form <a href="Komplexe_Konjugation" title="Komplexe Konjugation">komplex konjugierter</a> Paare im Gamma-Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \gamma (f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \gamma (f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e17da88ce7a2a9ecc819420f2c48ab6b0c195ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.475ex; height:2.843ex;" alt="{\displaystyle \textstyle \gamma (f,s)}" loading="lazy"> vor. Außerdem ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (\kappa _{j})>-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (\kappa _{j})>-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a64a108fad28f356227bc59877b46de0a7f1dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.051ex; height:3.009ex;" alt="{\displaystyle \textstyle \Re (\kappa _{j})>-1}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle j\in \{1,\ldots ,d\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle j\in \{1,\ldots ,d\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e26230584282c2f3b3dbcbdfb8f7282d38e68b7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:13.707ex; height:2.843ex;" alt="{\displaystyle \textstyle j\in \{1,\ldots ,d\}}" loading="lazy">. Diese letzte Bedingungen sorgt dafür, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \gamma (f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \gamma (f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e17da88ce7a2a9ecc819420f2c48ab6b0c195ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.475ex; height:2.843ex;" alt="{\displaystyle \textstyle \gamma (f,s)}" loading="lazy"> keine <a href="Nullstelle" title="Nullstelle">Nullstellen</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e1e46d489a93365a2da543a99b1a8174ec06bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \textstyle \mathbb {C} }" loading="lazy"> und keine <a href="Polstelle" title="Polstelle">Polstellen</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)\geq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f49dced22a8cdeceea1b4c53e18c37a6ac7b25e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)\geq 1}" loading="lazy"> besitzt. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c97012a41789c72b71118f54204adb7ee25b35d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle \textstyle \Re }" loading="lazy"> bezeichnet den <a href="Realteil" class="mw-redirect" title="Realteil">Realteil</a> einer komplexen Zahl.

B-4: Absolute Konvergenz der Dirichlet-Reihe und des Euler-Produkts

Sowohl die Dirichlet-Reihe als auch das Euler-Produkt, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet sind, konvergieren für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> <a href="Absolute_Konvergenz" class="mw-redirect" title="Absolute Konvergenz">absolut</a>.

B-5: Übereinstimmung von L-Funktion, Dirichlet-Reihe und Euler-Produkt in einer komplexen Halbebene

Die L-Funktion, die Dirichlet-Reihe und das Euler-Produkt, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet sind, stimmen in der komplexen Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> überein:

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(f,s)=\sum _{n\in \mathbb {N} }\lambda (f,n)n^{-s}=\prod _{p\in \mathbb {P} }(1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e955875c25a0579ccb85d709d81584ab5252631a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:75.877ex; height:5.843ex;" alt="{\displaystyle L(f,s)=\sum _{n\in \mathbb {N} }\lambda (f,n)n^{-s}=\prod _{p\in \mathbb {P} }(1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}.}" loading="lazy"></dd></dl>

B-6: Analytische Fortsetzbarkeit und Polstellen

Schon aus den Bedingungen, die die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnete Dirichlet-Reihe erfüllen muss, folgt die <a href="Holomorphe_Funktion" title="Holomorphe Funktion">Holomorphie</a> der vollständigen L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Lambda (f,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Lambda (f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d53082a4edec9c137a0146c375f1ee7381340baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.825ex; height:2.843ex;" alt="{\displaystyle \textstyle \Lambda (f,s)}" loading="lazy"> in der <a href="Halbebene" title="Halbebene">Halbebene</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy">. Diese muss aber auch <a href="Analytische_Fortsetzung" title="Analytische Fortsetzung">analytisch fortsetzbar</a> sein zu einer <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphen Funktion</a> der Ordnung 1 auf ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">, welche Polstellen höchstens in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle s=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b98b8e99360e0e6f889798d5b0d123f1cc1a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle s=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9939f5ed4234e51b6d1de83bac024687747f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=1}" loading="lazy"> besitzt.

B-7: Absolutbetrag der Wurzelzahl

Die Wurzelzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \epsilon (f)\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \epsilon (f)\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d7b6dead0fd7693579e330ea0ae5f0acc8deacc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.551ex; height:2.843ex;" alt="{\displaystyle \textstyle \epsilon (f)\in \mathbb {C} }" loading="lazy"> besitzt den <a href="Absolutbetrag" class="mw-redirect" title="Absolutbetrag">Absolutbetrag</a> 1. Also: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle |\epsilon (f)|=1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle |\epsilon (f)|=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e350cf92869b60ab47faefa88f68a39489c3048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.587ex; height:2.843ex;" alt="{\displaystyle \textstyle |\epsilon (f)|=1}" loading="lazy">.

B-8: Anforderungen an die Objekte, die dem Dual von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet sind

Was das Dual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> angeht, so muss gelten:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda ({\bar {f}},n)={\bar {\lambda }}(f,n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea78c78d3720d6d0fec6c131aa445cc396fb09a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.262ex; height:3.176ex;" alt="{\displaystyle \textstyle \lambda ({\bar {f}},n)={\bar {\lambda }}(f,n)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3579ec499388d52fefca601a35c16ab65ee6e4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle \textstyle n\in \mathbb {N} }" loading="lazy">, sowie

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \gamma ({\bar {f}},s)=\gamma (f,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0efbc91f412208326554cc65bc0a749919426df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.468ex; height:3.176ex;" alt="{\displaystyle \textstyle \gamma ({\bar {f}},s)=\gamma (f,s)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle q({\bar {f}})=q(f)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle q({\bar {f}})=q(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49d3fc714a204e2a4a1762b5c790f94a566efe5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.834ex; height:3.176ex;" alt="{\displaystyle \textstyle q({\bar {f}})=q(f)}" loading="lazy">. Das bedeutet: In der Dirichlet-Reihe, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> zugeordnet ist, sind die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2037b38f3c0b6632ea7e5f4f11547276687288d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \textstyle \lambda }" loading="lazy">-Koeffizienten gerade die komplex konjugierten Zahlen der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2037b38f3c0b6632ea7e5f4f11547276687288d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \textstyle \lambda }" loading="lazy">-Koeffizienten in der Dirichlet-Reihe, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> zugeordnet ist. Die Gamma-Faktoren und Führer, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> zugeordnet sind, stimmen überein.

B-9: Funktionalgleichung

Die beiden vollständigen L-Funktionen, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7da7b4fb34e6465d05809bf59a15932860c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}}" loading="lazy"> zugeordnet sind, erfüllen die Funktionalgleichung

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (f,s)=\epsilon (f)\Lambda ({\bar {f}},1-s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f8a15a4f85d7a363a828fa4a285710239d8e310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.204ex; height:3.176ex;" alt="{\displaystyle \Lambda (f,s)=\epsilon (f)\Lambda ({\bar {f}},1-s)}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle s\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f030d5fc9630ec99be496c46bc45c53c34033e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle \textstyle s\in \mathbb {C} }" loading="lazy">.

Der Definitionsansatz von Iwaniec und Kowalski spiegelt die Tatsache wider, dass eine Funktion, die als L-Funktion angesehen wird, typischerweise als Zuordnung der L-Funktion zu einem mathematischen Objekt (z. B. Dirichlet-Charakter, algebraischer Zahlkörper) auftritt. Ihr Definitionsansatz ist abstrakt und unvollständig, da er die Frage offen lässt, was denn jene mathematischen Objekte genau sind und wie jene Zuordnung stattzufinden hat.

Ohne Bezug zu anderen, mathematischen Objekten kommt der Definitionsansatz des norwegisch-US-amerikanischen Mathematikers <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> von 1989 aus. In einer nicht-abstrakten, eindeutigen Definition spezifiziert er eine Teilmenge der Menge aller Dirichlet-Reihen, deren Elemente bestimmte Eigenschaften erfüllen müssen: absolute Konvergenz der Dirichlet-Reihe, analytische Fortsetzbarkeit, Funktionalgleichung, <a href="Ramanujan-Vermutung" title="Ramanujan-Vermutung">Ramanujan-Vermutung</a><a href="#cite_note-4">[Anm. 1]</a> und Euler-Produkt. Diese Teilmenge wird heute als <a href="Selberg-Klasse" title="Selberg-Klasse">Selberg-Klasse</a> bezeichnet.<a href="#cite_note-5">[4]</a>

Die alles überragende Hypothese und der motivierende Hintergrund für die Definition der Selberg-Klasse ist die so genannte Große Riemannsche Vermutung. Auf die Selberg-Klasse angewandt besagt diese Vermutung: keine Nullstelle einer analytischen Fortsetzung einer Dirichlet-Reihe in der Selberg-Klasse besitzt einen Realteil größer als 1/2. Diese Vermutung entspricht im Fall des (vermeintlich) einfachsten Elements der Selberg-Klasse (Riemannsche Dirichlet-Reihe samt ihrer analytischen Fortsetzung zur Riemannschen Zeta-Funktion) der <a href="Riemannsche_Vermutung" title="Riemannsche Vermutung">Riemannschen Vermutung</a>, welche bis heute weder bewiesen noch widerlegt ist. Die Große Riemannsche Vermutung konnte bislang für kein einziges Element der Selberg-Klasse bewiesen oder widerlegt werden.

Vor diesem Hintergrund sind auch die noch existierenden Unzulänglichkeiten bei der Definition des Begriffs „L-Funktion“ zu sehen: man möchte den Begriff „L-Funktion“ so definieren, dass L-Funktionen die Große Riemannsche Vermutung beweisbar erfüllen – andererseits konnte man bislang noch nicht einmal den einfachsten Fall (Riemannsche Vermutung für die Riemannsche Zeta-Funktion) beweisen, was ein Zeichen für mangelndes Verständnis der Riemannschen Zeta-Funktion sein könnte und damit eine eindeutige Definition des verallgemeinernden Begriffs der „L-Funktion“ erschwert.

<div class="mw-heading mw-heading2"><h2 id="Beispiele_von_L-Funktionen">Beispiele von L-Funktionen</h2></div>

Dieser Abschnitt gibt einen Überblick über grundlegende Beispiele von L-Funktionen.

<div class="mw-heading mw-heading3"><h3 id="Riemannsche_Zeta-Funktion">Riemannsche Zeta-Funktion</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannsche Zeta-Funktion</a></div>

Das einfachste Beispiel einer L-Funktion und gleichzeitig Ausgangspunkt für jede Definition des Begriffs „L-Funktion“ ist die Riemannsche Zeta-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \zeta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898b9bce03a533d123c225d5f5ac48635e52ecab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \textstyle \zeta }" loading="lazy">.<a href="#cite_note-6">[5]</a>

Eines der möglichen „arithmetischen Objekte“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1ac73cc984d65a6529c72ac4b63642525b658dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle \textstyle f}" loading="lazy"> im Sinne des Definitionsansatzes von Iwaniec und Kowalski, welchem diese L-Funktion zugeordnet werden kann, ist der <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d74881d420116e547703edb7942c6448853d6ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \textstyle \mathbb {Q} }" loading="lazy"> der <a href="Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a>. Ihre Dirichlet-Reihe

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {N} }{\frac {1}{n^{s}}}\,,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8494275066b5f6df48635628d839b99efe76c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.01ex; height:6.509ex;" alt="{\displaystyle \sum _{n\in \mathbb {N} }{\frac {1}{n^{s}}}\,,}" loading="lazy"></dd></dl>

also

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (\mathbb {Q} ,n)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dbb08cfe20566a2bdb40b701091cb5be26a0361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.662ex; height:2.843ex;" alt="{\displaystyle \lambda (\mathbb {Q} ,n)=1}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3579ec499388d52fefca601a35c16ab65ee6e4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle \textstyle n\in \mathbb {N} }" loading="lazy">, konvergiert für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> absolut. Zusammen mit ihrem ebenfalls absolut konvergenten Euler-Produkt gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy">:<a href="#cite_note-7">[6]</a>

</mrow>

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta (s)=L(\mathbb {Q} ,s)=\sum _{n\in \mathbb {N} }{\frac {1}{n^{s}}}=\prod _{p\in \mathbb {P} }(1-p^{-s})^{-1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7d06931c331a2dc899adfaced751a2169c3b4ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.804ex; height:6.676ex;" alt="{\displaystyle \zeta (s)=L(\mathbb {Q} ,s)=\sum _{n\in \mathbb {N} }{\frac {1}{n^{s}}}=\prod _{p\in \mathbb {P} }(1-p^{-s})^{-1}.}" loading="lazy"></dd></dl>

Da alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (\mathbb {Q} ,n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (\mathbb {Q} ,n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd1e164018faa7fccbb969fbafbe13b48ff3ad4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.401ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (\mathbb {Q} ,n)}" loading="lazy"> reell sind, nämlich gleich 1, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \zeta (s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \zeta (s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecb7531c39821e764de019ae0e11fa5e330040f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.995ex; height:2.843ex;" alt="{\displaystyle \textstyle \zeta (s)}" loading="lazy"> selbstdual. Das zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f=\mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f=\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/651fb4ad147e64bf2878e9c968a9cb373a680f3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.185ex; height:2.509ex;" alt="{\displaystyle \textstyle f=\mathbb {Q} }" loading="lazy"> duale Objekt ist also ebenfalls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d74881d420116e547703edb7942c6448853d6ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \textstyle \mathbb {Q} }" loading="lazy">, somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\bar {f}}=\mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\bar {f}}=\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/500e418992a9d750aaa545d52f5d082cc0525628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.606ex; height:3.009ex;" alt="{\displaystyle \textstyle {\bar {f}}=\mathbb {Q} }" loading="lazy">.

Der Grad des Euler-Produktes der Riemannschen Zeta-Funktion ist

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<annotation encoding="application/x-tex">{\displaystyle d=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958b426c5ace91d4fb7f5a3becd7b21dba288d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=1}" loading="lazy">.</dd></dl>

Für ihre lokalen Parameter bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy"> gilt:

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<annotation encoding="application/x-tex">{\displaystyle \alpha (\mathbb {Q} ,p)=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2a526111233487229780126f15926cab86fe2f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.569ex; height:2.843ex;" alt="{\displaystyle \alpha (\mathbb {Q} ,p)=1}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p\in \mathbb {P} }">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p\in \mathbb {P} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea713a60b8d705dc0bf0d79ec5d0944f88eed5aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle \textstyle p\in \mathbb {P} }" loading="lazy">.

Üblicherweise wird für die Riemannsche Zeta-Funktion der folgende Gamma-Faktor verwendet:

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<annotation encoding="application/x-tex">{\displaystyle \gamma (\mathbb {Q} ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02967f0420dfe9a2daa32d5424354dcf29449cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.486ex; height:4.843ex;" alt="{\displaystyle \gamma (\mathbb {Q} ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right).}" loading="lazy"></dd></dl>

Der lokale Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a67f7cbf12eeb5f34974990a76e9b1f6d7e237a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \textstyle \kappa }" loading="lazy"> im Unendlichen ist dann also 0.

Der Führer von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \zeta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898b9bce03a533d123c225d5f5ac48635e52ecab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \textstyle \zeta }" loading="lazy"> ist

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<annotation encoding="application/x-tex">{\displaystyle \textstyle q(\mathbb {Q} )=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/406943387987aabcc73fb343a1fe64ba907891e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.948ex; height:2.843ex;" alt="{\displaystyle \textstyle q(\mathbb {Q} )=1}" loading="lazy">,</dd></dl>

so dass die vollständige Riemannsche Zeta-Funktion die Gestalt

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<msup>

<mfrac>

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</msup>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\mathbb {Q} ,s):=\gamma (\mathbb {Q} ,s)L(\mathbb {Q} ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right)\zeta (s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4816bccef5444237227498db1ebc6e2d8dd1b448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.259ex; height:4.843ex;" alt="{\displaystyle \Lambda (\mathbb {Q} ,s):=\gamma (\mathbb {Q} ,s)L(\mathbb {Q} ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right)\zeta (s)}" loading="lazy"></dd></dl>

annimmt. Diese Definition ist nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> gültig, da nur in dieser Halbebene die Riemannsche Zeta-Funktion über ihre Dirichlet-Reihe oder ihr Euler-Produkt definiert werden kann. Allerdings besitzt die vollständige Riemannsche Zeta-Funktion eine analytische Fortsetzung zu einer meromorphen Funktion auf der ganzen komplexen Zahlenebene. Diese Fortsetzung ist holomorph bis auf zwei einfache Polstellen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=0}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle s=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24b98b8e99360e0e6f889798d5b0d123f1cc1a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=1}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle s=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9939f5ed4234e51b6d1de83bac024687747f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=1}" loading="lazy"> mit den <a href="Residuum_(Funktionentheorie)" title="Residuum (Funktionentheorie)">Residuen</a> −1 bzw. 1.<a href="#cite_note-8">[7]</a>

Bezeichnet man auch die fortgesetzte, vollständige Riemannsche Zeta-Funktion mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3019d0254b7e41e818fefe38e5ef865a8993f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \textstyle \Lambda }" loading="lazy">, so erfüllt sie mit der Wurzelzahl

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon (\mathbb {Q} )=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3db55b2e3fc15024d78dace873e2fb901523b2df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.822ex; height:2.843ex;" alt="{\displaystyle \epsilon (\mathbb {Q} )=1}" loading="lazy"></dd></dl>

die Funktionalgleichung<a href="#cite_note-9">[8]</a>

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\mathbb {Q} ,s)=\Lambda (\mathbb {Q} ,1-s).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/198f16a109b5b5262500f690e233e9a0a10dff90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.458ex; height:2.843ex;" alt="{\displaystyle \Lambda (\mathbb {Q} ,s)=\Lambda (\mathbb {Q} ,1-s).}" loading="lazy"></dd></dl>

Damit besitzt auch die zunächst nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> durch ihre Dirichlet-Reihe oder Euler-Produkt definierte Riemannsche Zeta-Funktion eine analytische Fortsetzung zu einer meromorphen Funktion auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">, welche einzig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle s=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9939f5ed4234e51b6d1de83bac024687747f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=1}" loading="lazy"> nicht definiert ist, da sie dort über eine einfache Polstelle mit Residuum 1 verfügt. Behält man die Bezeichnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \zeta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898b9bce03a533d123c225d5f5ac48635e52ecab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \textstyle \zeta }" loading="lazy"> auch für die fortgesetzte Riemannsche Zeta-Funktion bei, so erfüllt sie die Funktionalgleichung<a href="#cite_note-10">[9]</a>

<msup>

<mfrac>

</mfrac>

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</msup>

<mrow>

<mfrac>

</mfrac>

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<msup>

<mfrac>

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</mfrac>

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</msup>

<mrow>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\pi ^{-{\frac {1-s}{2}}}\,\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fadb375cfc6c13055019040520d40333bee0391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.452ex; height:6.176ex;" alt="{\displaystyle \pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\pi ^{-{\frac {1-s}{2}}}\,\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s).}" loading="lazy"></dd></dl>

Die (analytisch fortgesetzte) Riemannsche Zeta-Funktion birgt eine der wichtigsten Fragen der analytischen Zahlentheorie, nämlich die Frage nach der genauen Lage ihrer sogenannten nicht-trivialen Nullstellen. Diese liegen im kritischen Streifen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 0<\Re (s)<1}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle 0<\Re (s)<1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87ef5644a94513c5a9ebdd72c5d240347f932ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.346ex; height:2.843ex;" alt="{\displaystyle \textstyle 0<\Re (s)<1}" loading="lazy">. Die <a href="Riemannsche_Vermutung" title="Riemannsche Vermutung">Riemannsche Vermutung</a> aus dem Jahr 1859 – bis heute weder bewiesen noch widerlegt – stellt die These auf, alle nicht-trivialen Nullstellen der Riemannschen Zeta-Funktion besäßen den Realteil 1/2. Ein Beweis dieser Vermutung würde besonders gute Abschätzungen über die Verteilung der Primzahlen gestatten.

<div class="mw-heading mw-heading3"><h3 id="Dirichletsche_L-Funktionen">Dirichletsche L-Funktionen</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Dirichletsche_L-Funktion" title="Dirichletsche L-Funktion">Dirichletsche L-Funktion</a></div>

Die nächsten Verwandten der Riemannschen Zeta-Funktion sind die Dirichletschen L-Funktionen, welche die Riemannsche Zeta-Funktion als Spezialfall enthalten. Sind in der zur Riemannschen Zeta-Funktion gehörenden Dirichlet-Reihe noch alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">- Koeffizienten gleich 1, so werden diese bei Dirichletschen L-Funktionen mit Hilfe eines <a href="Dirichlet-Charakter" class="mw-redirect" title="Dirichlet-Charakter">Dirichlet-Charakters</a> definiert. Sie nehmen somit komplexe Werte mit dem Absolutbetrag 1 an oder sind gleich 0. Sei also für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m\in \mathbb {N} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle m\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68162a3d51e9be369e3b8b39bdefe5659fb8e569.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle \textstyle m\in \mathbb {N} }" loading="lazy"> ein Dirichlet-Charakter modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy">

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi :(\mathbb {Z} /m)^{\times }\longrightarrow S^{1}:=\{z\in \mathbb {C} :|z|=1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10f53f794e8b9ea946163730422e10095b6edc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.394ex; height:3.176ex;" alt="{\displaystyle \chi :(\mathbb {Z} /m)^{\times }\longrightarrow S^{1}:=\{z\in \mathbb {C} :|z|=1\}}" loading="lazy"></dd></dl>

gegeben, d. h. ein <a href="Gruppenhomomorphismus" title="Gruppenhomomorphismus">Gruppenhomomorphismus</a> von der <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppe</a> der bzgl. der Multiplikation <a href="Inverses_Element" title="Inverses Element">invertierbaren Elemente</a> des <a href="Restklassenring" title="Restklassenring">Restklassenrings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Z} /m}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Z} /m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50196cdf2002dbbd3a65144d1c3aa52d86699612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.753ex; height:2.843ex;" alt="{\displaystyle \textstyle \mathbb {Z} /m}" loading="lazy"> in die <a href="Kreisgruppe" title="Kreisgruppe">Kreisgruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle S^{1}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba2f621a5d1d837f3e1a962114bd87d808b6599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle \textstyle S^{1}}" loading="lazy"> der komplexen Zahlen mit Absolutbetrag 1. Ein solcher Dirichlet-Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> heißt primitiv und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy"> der Führer von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy">, wenn er nicht schon durch eine Komposition

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mover>

</mrow>

<msup>

</msup>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /m)^{\times }\longrightarrow (\mathbb {Z} /m')^{\times }\;{\stackrel {\chi '}{\longrightarrow }}\;S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38cb45ce7dac39ebce7c6881039ed6ce3baa01ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.599ex; height:4.509ex;" alt="{\displaystyle (\mathbb {Z} /m)^{\times }\longrightarrow (\mathbb {Z} /m')^{\times }\;{\stackrel {\chi '}{\longrightarrow }}\;S^{1}}" loading="lazy"></dd></dl>

aus einem Dirichlet-Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi '}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c6a616a08b128edeb8c24d50735e362fcb09ee8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.14ex; height:2.676ex;" alt="{\displaystyle \textstyle \chi '}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec1a7c4693776a926ef4e38d709c0d13ae68db0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.725ex; height:2.343ex;" alt="{\displaystyle \textstyle m'}" loading="lazy"> mit einem echten Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec1a7c4693776a926ef4e38d709c0d13ae68db0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.725ex; height:2.343ex;" alt="{\displaystyle \textstyle m'}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy"> hervorgeht. Mit Hilfe eines solchen Dirichlet-Charakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> definiert man die nachfolgende Abbildung, welche ebenfalls mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> und als Dirichlet-Charakter modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy"> bezeichnet wird:<a href="#cite_note-11">[10]</a>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :{\text{ }}\mathbb {Z} \to \mathbb {C} ,{\text{ }}\chi (n)={\begin{cases}\chi (n\operatorname {mod} m)&{\text{falls}}\quad \operatorname {ggT} (n,m)=1\\0&{\text{falls}}\quad \operatorname {ggT} (n,m)>1.\end{cases}}}">

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :{\text{ }}\mathbb {Z} \to \mathbb {C} ,{\text{ }}\chi (n)={\begin{cases}\chi (n\operatorname {mod} m)&{\text{falls}}\quad \operatorname {ggT} (n,m)=1\\0&{\text{falls}}\quad \operatorname {ggT} (n,m)>1.\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c8e2b7d47a1558e275eb392549eeae074161430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.852ex; height:6.176ex;" alt="{\displaystyle \chi :{\text{ }}\mathbb {Z} \to \mathbb {C} ,{\text{ }}\chi (n)={\begin{cases}\chi (n\operatorname {mod} m)&{\text{falls}}\quad \operatorname {ggT} (n,m)=1\\0&{\text{falls}}\quad \operatorname {ggT} (n,m)>1.\end{cases}}}" loading="lazy"></dd></dl>

Die trivialen Dirichlet-Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi ^{0}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9f1687909a372c16e88a2cef22afd17ce8a8224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.843ex;" alt="{\displaystyle \textstyle \chi ^{0}}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> besitzen den Funktionswert 1, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \operatorname {ggT} (n,m)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \operatorname {ggT} (n,m)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/426a62562309f54731100e961d4e4a1f78624268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.542ex; height:2.843ex;" alt="{\displaystyle \textstyle \operatorname {ggT} (n,m)=1}" loading="lazy">, andernfalls 0. Der triviale Dirichlet-Charakter modulo 1 heißt der Hauptcharakter. Er erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (n)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi (n)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/288e5eac6ec07761fbd69dd0f2e628258be61938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.92ex; height:2.843ex;" alt="{\displaystyle \chi (n)=1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">.

Ist nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi :\mathbb {Z} \to \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi :\mathbb {Z} \to \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78f98960a0bf4fb1795dbf0c724721b3a633c5ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.235ex; height:2.509ex;" alt="{\displaystyle \textstyle \chi :\mathbb {Z} \to \mathbb {C} }" loading="lazy"> ein primitiver Dirichlet-Charakter modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy">, so ordnet man diesem arithmetischen Objekt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> folgendermaßen eine L-Funktion zu: Mit

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (\chi ,n):=\chi (n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9caa1e6bca6b1b5bdb362cbaeac0e33830c3597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.453ex; height:2.843ex;" alt="{\displaystyle \lambda (\chi ,n):=\chi (n)}" loading="lazy"></dd></dl>

konvergiert die Dirichlet-Reihe (auch Dirichletsche L-Reihe genannt)

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s):=\sum _{n\in \mathbb {N} }{\frac {\lambda (\chi ,n)}{n^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9c76e9489a5880c6d647c827a767825c15cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.679ex; height:7.009ex;" alt="{\displaystyle L(\chi ,s):=\sum _{n\in \mathbb {N} }{\frac {\lambda (\chi ,n)}{n^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}}}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> absolut.<a href="#cite_note-13">[12]</a>

Mit den lokalen Parametern bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b99f99e3ca9227614414c838c629a136400a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle \textstyle p}" loading="lazy">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha (\chi ,p):=\chi (p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a13cbd6ec17b564da466683d7979c642c1c30eb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.135ex; height:2.843ex;" alt="{\displaystyle \alpha (\chi ,p):=\chi (p)}" loading="lazy"></dd></dl>

gilt dies auch für das zugehörende Euler-Produkt, und man hat die Identität<a href="#cite_note-14">[13]</a>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)=\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}}=\prod _{p\in \mathbb {P} }(1-\chi (p)p^{-s})^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1dbbf98677cb9b9ccfd687bff8db1bf78a3a5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.405ex; height:7.176ex;" alt="{\displaystyle L(\chi ,s)=\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}}=\prod _{p\in \mathbb {P} }(1-\chi (p)p^{-s})^{-1}}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy">. Wie bei der Riemannschen Zeta-Funktion ist

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958b426c5ace91d4fb7f5a3becd7b21dba288d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=1}" loading="lazy"></dd></dl>

der Grad des Euler-Produkts. Setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf3d57ae17916d4680cecf9448318a7884262937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle \textstyle \kappa =0}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi (-1)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi (-1)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b75d4771b9ffa43187d1fa111ebcf8ab48fec27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.496ex; height:2.843ex;" alt="{\displaystyle \textstyle \chi (-1)=1}" loading="lazy"> (in diesem Fall heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> gerade), und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa =1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a96023929dd5140413670a6ba882b5d023aac966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle \textstyle \kappa =1}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi (-1)=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi (-1)=-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/092f829f3713955ea26f4e25d3d3891d407594c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.304ex; height:2.843ex;" alt="{\displaystyle \textstyle \chi (-1)=-1}" loading="lazy"> (in diesem Fall heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> ungerade), so ist

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (\chi ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s+\kappa }{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a57dba30a9f90be7237b94d1fafa20dce5fc6128.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.852ex; height:6.176ex;" alt="{\displaystyle \gamma (\chi ,s)=\pi ^{-{\frac {s}{2}}}\,\Gamma \left({\frac {s+\kappa }{2}}\right)}" loading="lazy"></dd></dl>

der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> zugeordnete Gamma-Faktor. Jenes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \kappa \in \{0,1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \kappa \in \{0,1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85bed67554e36a8bec896d47b56e1ba4871d349d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.863ex; height:2.843ex;" alt="{\displaystyle \textstyle \kappa \in \{0,1\}}" loading="lazy"> ist also der lokale Parameter an der unendlichen Primstelle.

Der Führer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa35069f79eaab8d7586d629e6a9e4d2906cfe3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle \textstyle m}" loading="lazy"> des primitiven Dirichlet-Charakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> ist auch der Führer der Dirichletschen L-Funktion:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(\chi )=m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b57c53168817967f69fcfd363afb6c59f8e5f4e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.473ex; height:2.843ex;" alt="{\displaystyle q(\chi )=m}" loading="lazy">.</dd></dl>

Die vollständige Dirichletsche L-Funktion besitzt somit die Form<a href="#cite_note-15">[14]</a>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s):=q(\chi )^{\frac {s}{2}}\gamma (\chi ,s)L(\chi ,s)=\left({\frac {m}{\pi }}\right)^{\frac {s}{2}}\Gamma \left({\frac {s+\kappa }{2}}\right)\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b65d14035cc3ace7bd5c0353d2f05fbe68d39cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.564ex; height:7.009ex;" alt="{\displaystyle \Lambda (\chi ,s):=q(\chi )^{\frac {s}{2}}\gamma (\chi ,s)L(\chi ,s)=\left({\frac {m}{\pi }}\right)^{\frac {s}{2}}\Gamma \left({\frac {s+\kappa }{2}}\right)\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}},}" loading="lazy"></dd></dl>

eine Definition, die nur für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb9ca52e28f39766831aaa9400febff20a68546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \textstyle \Re (s)>1}" loading="lazy"> gilt, da nur dort die verwendete Dirichlet-Reihe konvergiert. Eine solche vollständige Dirichletsche L-Funktion kann aber analytisch auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e1e46d489a93365a2da543a99b1a8174ec06bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \textstyle \mathbb {C} }" loading="lazy"> fortgesetzt werden. Dabei entsteht eine <a href="Ganze_Funktion" title="Ganze Funktion">ganze Funktion</a>, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> ein nicht-trivialer Dirichlet-Charakter ist.<a href="#cite_note-16">[15]</a>

Andernfalls hat die fortgesetzte Funktion einen einfachen Pol in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle s=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle s=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9939f5ed4234e51b6d1de83bac024687747f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \textstyle s=1}" loading="lazy"> mit Residuum 1.<a href="#cite_note-17">[16]</a> Das zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> duale Objekt ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\chi }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\chi }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69929a325e127f2929281960e4716e1b760e4114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.57ex; height:2.676ex;" alt="{\displaystyle {\overline {\chi }}}" loading="lazy">, also derjenige Dirichlet-Charakter, der aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> durch komplexe Konjugation der Funktionswerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy"> hervorgeht, d. h.

<mover>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\overline {\chi }}(n)={\overline {\chi (n)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca992dc65d3a901dfbdf34d6eda14edae4b9bde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.647ex; height:3.676ex;" alt="{\displaystyle \textstyle {\overline {\chi }}(n)={\overline {\chi (n)}}}" loading="lazy"></dd></dl>

für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3579ec499388d52fefca601a35c16ab65ee6e4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle \textstyle n\in \mathbb {N} }" loading="lazy">. Die Wurzelzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \epsilon (\chi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \epsilon (\chi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b77aaa0f1c964373f7c322a5590cf17e04faecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle \textstyle \epsilon (\chi )}" loading="lazy"> kann mit Hilfe der <a href="Gau%C3%9Fsche_Summe" title="Gaußsche Summe">Gaußschen Summe</a><a href="#cite_note-18">[17]</a>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau (\chi )=\sum _{x\operatorname {mod} q(\chi )}\chi (x)\exp(2\pi ix/q(\chi ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41bc8676d234c12624bb5dd035d7e490b5fc8148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:36.302ex; height:6.009ex;" alt="{\displaystyle \tau (\chi )=\sum _{x\operatorname {mod} q(\chi )}\chi (x)\exp(2\pi ix/q(\chi ))}" loading="lazy"></dd></dl>

berechnet werden, in der sich die Summation über alle Restklassen modulo des Führers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle q(\chi )=m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle q(\chi )=m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6dc81ea2d719441ca535e194f29025e95c92b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.473ex; height:2.843ex;" alt="{\displaystyle \textstyle q(\chi )=m}" loading="lazy"> erstreckt sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f3ac3a54990c3e941060e44a8850d1c5824f537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \textstyle \pi }" loading="lazy"> die <a href="Kreiszahl" title="Kreiszahl">Kreiszahl</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle i}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b69dce609a74290821dcbd7db868b1af9c061fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle \textstyle i}" loading="lazy"> die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1185b570f67b4221307626254f64f9e619e769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }" loading="lazy"> die <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> bezeichnen. Mit

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon (\chi )={\begin{cases}{\frac {\tau (\chi )}{\sqrt {q(\chi )}}}&{\text{falls}}\quad \chi (-1)=1\\{\frac {\tau (\chi )}{i{\sqrt {q(\chi )}}}}&{\text{falls}}\quad \chi (-1)=-1\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c1de4e83bedea4227d5eb3dd55aff013c0f8882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:37.269ex; height:10.176ex;" alt="{\displaystyle \epsilon (\chi )={\begin{cases}{\frac {\tau (\chi )}{\sqrt {q(\chi )}}}&{\text{falls}}\quad \chi (-1)=1\\{\frac {\tau (\chi )}{i{\sqrt {q(\chi )}}}}&{\text{falls}}\quad \chi (-1)=-1\end{cases}}}" loading="lazy"></dd></dl>

erfüllt dann die fortgesetzte, vollständige Dirichletsche L-Funktion die Funktionalgleichung<a href="#cite_note-19">[18]</a>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)=\epsilon (\chi )\Lambda ({\overline {\chi }},1-s).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e52a186196c93c425d7c6a50da871991e54cff5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.075ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)=\epsilon (\chi )\Lambda ({\overline {\chi }},1-s).}" loading="lazy"></dd></dl>

Wie von Wurzelzahlen gefordert, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle |\epsilon (\chi )|=1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle |\epsilon (\chi )|=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42d04f3863b2bec30bd922db0a031a0c9578a83a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.763ex; height:2.843ex;" alt="{\displaystyle \textstyle |\epsilon (\chi )|=1}" loading="lazy">, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle |\tau (\chi )|={\sqrt {q(\chi )}}}">

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle |\tau (\chi )|={\sqrt {q(\chi )}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08452ca3806d9ca1dc34bca31882ae254318eb54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.516ex; height:3.343ex;" alt="{\displaystyle \textstyle |\tau (\chi )|={\sqrt {q(\chi )}}}" loading="lazy">.<a href="#cite_note-20">[19]</a>

Die Dirichletschen L-Funktionen umfassen die Riemannsche Zeta-Funktion, da diese aus dem trivialen Dirichlet-Charakter modulo 1, also dem Hauptcharakter, entsteht.<a href="#cite_note-21">[20]</a>

Der deutsche Mathematiker Peter Gustav Dirichlet verwendete 1837 die nach ihm benannten Dirichletschen L-Funktionen, um den <a href="Dirichletscher_Primzahlsatz" class="mw-redirect" title="Dirichletscher Primzahlsatz">Dirichletschen Primzahlsatz</a> zu beweisen, wonach in jeder <a href="Arithmetische_Folge" title="Arithmetische Folge">arithmetischen Folge</a> (auch arithmetische Progression genannt)

</mrow>

<mtext> wobei </mtext>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,a\pm n,a\pm 2n,a\pm 3n,\ldots ,{\text{ mit }}\operatorname {ggT} (a,n)=1,{\text{ wobei }}a,n\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fc2be0c931604362a8f7c13ea4161d8e7a5f668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.828ex; height:2.843ex;" alt="{\displaystyle a,a\pm n,a\pm 2n,a\pm 3n,\ldots ,{\text{ mit }}\operatorname {ggT} (a,n)=1,{\text{ wobei }}a,n\in \mathbb {N} ,}" loading="lazy"></dd></dl>

d. h. in jeder Restklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a\operatorname {mod} n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle a\operatorname {mod} n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1628a6d66902de63517fcb680a3cce4c619adc87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.79ex; height:2.176ex;" alt="{\displaystyle \textstyle a\operatorname {mod} n}" loading="lazy">, unendlich viele Primzahlen liegen.<a href="#cite_note-22">[21]</a>

Das entscheidende Argument im Beweis des Dirichletschen Primzahlsatzes ist die Erkenntnis, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Lambda (\chi ,1)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Lambda (\chi ,1)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46fe48de129c59d0b20dc0f0b82263278c1bbc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.335ex; height:2.843ex;" alt="{\displaystyle \textstyle \Lambda (\chi ,1)\neq 0}" loading="lazy"> gilt für jeden nicht-trivialen Dirichlet-Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f4a0e9fa17ff37f01dbcae126a04465de07893d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \textstyle \chi }" loading="lazy">.<a href="#cite_note-24">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Dedekindsche_L-Funktionen">Dedekindsche L-Funktionen</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Dedekindsche_Zeta-Funktion" title="Dedekindsche Zeta-Funktion">Dedekindsche Zeta-Funktion</a></div>

Die Riemannsche Zeta-Funktion bezieht sich auf den Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d74881d420116e547703edb7942c6448853d6ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \textstyle \mathbb {Q} }" loading="lazy"> der rationalen Zahlen, dem einfachsten algebraischen Zahlkörper. Dedekindsche L-Funktionen verallgemeinern diesen Bezug auf beliebige algebraische Zahlkörper, also endlichen <a href="K%C3%B6rpererweiterung" title="Körpererweiterung">Körpererweiterungen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d74881d420116e547703edb7942c6448853d6ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \textstyle \mathbb {Q} }" loading="lazy"> wie zum Beispiel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} ({\sqrt[{3}]{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} ({\sqrt[{3}]{2}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e5f15d9da5fba1cba6b046c80fa6dadcbdc86f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:3.176ex;" alt="{\displaystyle \textstyle \mathbb {Q} ({\sqrt[{3}]{2}})}" loading="lazy">. Sei also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein algebraischer Zahlkörper und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1644286516fc1fb7ec8de773b43c72e5326b82b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.81ex; height:2.843ex;" alt="{\displaystyle n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }" loading="lazy"> sein <a href="Erweiterungsgrad" class="mw-redirect" title="Erweiterungsgrad">Erweiterungsgrad</a> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> sein <a href="Ganzheitsring" title="Ganzheitsring">Ganzheitsring</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{K}\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle d_{K}\in \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dbaa556e6c6ed93bed2f67ec3456f8ae415dec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.293ex; height:2.509ex;" alt="{\displaystyle d_{K}\in \mathbb {Z} }" loading="lazy"> seine <a href="Diskriminante_(algebraische_Zahlentheorie)" title="Diskriminante (algebraische Zahlentheorie)">Diskriminante</a>. Weiter seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}\in \mathbb {N} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{1}\in \mathbb {N} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/149eba5f572d6eb0872858881f4249a700a47b13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.022ex; height:2.509ex;" alt="{\displaystyle n_{1}\in \mathbb {N} _{0}}" loading="lazy"> die Anzahl der reellen <a href="Einbettung_(Mathematik)" title="Einbettung (Mathematik)">Einbettungen</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}\in \mathbb {N} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{2}\in \mathbb {N} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d475e905e493fcb5793dec19971fa8ad024cf46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.022ex; height:2.509ex;" alt="{\displaystyle n_{2}\in \mathbb {N} _{0}}" loading="lazy"> die Anzahl der Paare komplexer Einbettungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Es ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{K}=n_{1}+2n_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{K}=n_{1}+2n_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3fdf0e2a944f2577a44c81ec0dea7835fd7c0a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.087ex; height:2.509ex;" alt="{\displaystyle n_{K}=n_{1}+2n_{2}}" loading="lazy">.

Die Dedekindsche L-Funktion (auch <a href="Dedekindsche_Zeta-Funktion" title="Dedekindsche Zeta-Funktion">Dedekindsche Zeta-Funktion</a> genannt) bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> definiert durch<a href="#cite_note-25">[24]</a>

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<annotation encoding="application/x-tex">{\displaystyle \zeta _{K}(s):=L(K,s):=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e97aea2acf264a9d12ef799baaaafd7fc773400.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.664ex; height:6.843ex;" alt="{\displaystyle \zeta _{K}(s):=L(K,s):=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}" loading="lazy"></dd></dl>

In der Summe durchläuft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> alle vom <a href="Nullideal" class="mw-redirect" title="Nullideal">Nullideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"> verschiedenen, ganzen <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideale</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa778b8e4af4159a88c06b9a8f7d07aee59cdc78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:9.827ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }" loading="lazy"> bezeichnet die <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Absolutnorm</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">. Die Koeffizienten der Dirichlet-Reihe

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<annotation encoding="application/x-tex">{\displaystyle \sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\lambda (K,n)}{n^{s}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fbd1da79ec52e660e4c6ab2dbd0c0f4af1d0e69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.668ex; height:7.343ex;" alt="{\displaystyle \sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\lambda (K,n)}{n^{s}}}}" loading="lazy"></dd></dl>

sind also<a href="#cite_note-26">[25]</a>

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<annotation encoding="application/x-tex">{\displaystyle \lambda (K,n)=\#\{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}\mid {\mathcal {N}}({\mathfrak {a}})=n\}\in \mathbb {N} _{0}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d43045486cd85ad184fca732cdb0dd30b56ea917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.98ex; height:3.009ex;" alt="{\displaystyle \lambda (K,n)=\#\{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}\mid {\mathcal {N}}({\mathfrak {a}})=n\}\in \mathbb {N} _{0}.}" loading="lazy"></dd></dl>

Sie geben zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> die Anzahl der ganzen Ideale von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> mit Absolutnorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> an. Insbesondere sind alle Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (K,n)}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda (K,n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2568a271d88de04561b57e0794e5d9957fe79dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.659ex; height:2.843ex;" alt="{\displaystyle \lambda (K,n)}" loading="lazy"> reell und deshalb <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

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<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy"> selbstdual. Jene Dirichlet-Reihe konvergiert für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> absolut, ebenso wie das zugehörende Euler-Produkt

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<annotation encoding="application/x-tex">{\displaystyle \prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d485056f63a0a119d7a6c48279ada841cea5eb87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.261ex; height:6.843ex;" alt="{\displaystyle \prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}.}" loading="lazy"></dd></dl>

Dabei erstreckt sich das Produkt über alle vom Nullideal verschiedenen <a href="Primideal" title="Primideal">Primideale</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy">. Es gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> die Identität<a href="#cite_note-27">[26]</a>

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<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle L(K,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\lambda (K,n)}{n^{s}}}=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b5c73517c3e39e0dfe3835c277c009723e89804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:63.709ex; height:7.343ex;" alt="{\displaystyle L(K,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\sum _{n\in \mathbb {N} }{\frac {\lambda (K,n)}{n^{s}}}=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}.}" loading="lazy"></dd></dl>

Diese Gestalt des Euler-Produkts zeigt noch nicht die einzelnen Euler-Faktoren

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<annotation encoding="application/x-tex">{\displaystyle (1-\alpha _{1}(K,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(K,p)p^{-s})^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0a53ed5b8b733393e2a49b9d24ec3648fe97fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.553ex; height:3.176ex;" alt="{\displaystyle (1-\alpha _{1}(K,p)p^{-s})^{-1}\cdot \ldots \cdot (1-\alpha _{d}(K,p)p^{-s})^{-1}}" loading="lazy">.

Der Grad des Euler-Produkts ist jedenfalls gleich dem Grad der Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce6782a2935122dafec2947d1d7ba168be4399ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle d=[K:\mathbb {Q} ]=n_{K}=n_{1}+2n_{2}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2316c7890e43bce5b416f2d6c4e23e222156e3ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.252ex; height:2.843ex;" alt="{\displaystyle d=[K:\mathbb {Q} ]=n_{K}=n_{1}+2n_{2}.}" loading="lazy"></dd></dl>

Die lokalen Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}(K,p)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}(K,p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d73795bf270c7a0dced23c058626934a646527b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.476ex; height:3.009ex;" alt="{\displaystyle \alpha _{j}(K,p)}" loading="lazy"> hängen vom Zerlegungsverhalten der Ideale

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<annotation encoding="application/x-tex">{\displaystyle (p):=p{\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a27f3e63bc4cdd4c6b9d13b98e8286d0e4c1c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.436ex; height:2.843ex;" alt="{\displaystyle (p):=p{\mathcal {O}}_{K}}" loading="lazy">

ab: jedes Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)}">

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<annotation encoding="application/x-tex">{\displaystyle (p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b43fe924fbe82623de1aa506862fc522c5c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle (p)}" loading="lazy"> besitzt eine, bis auf die Reihenfolge der Faktoren, eindeutige Primidealzerlegung

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<annotation encoding="application/x-tex">{\displaystyle (p)=\prod _{\mathfrak {p}}{\mathfrak {p}}^{e_{\mathfrak {p}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c04170c7c9cc254169851c42aa5d5d62993fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.426ex; height:5.843ex;" alt="{\displaystyle (p)=\prod _{\mathfrak {p}}{\mathfrak {p}}^{e_{\mathfrak {p}}}}" loading="lazy"></dd></dl>

in Primideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/113fc5e920cf4f6f2b8965498d028a71f7f469c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.065ex; height:2.676ex;" alt="{\displaystyle 0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}" loading="lazy">, in der gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mathfrak {p}}\in \mathbb {N} _{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle e_{\mathfrak {p}}\in \mathbb {N} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ab450c0cb0cf25a0cd504fc76eced05f2fde5d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.711ex; height:2.843ex;" alt="{\displaystyle e_{\mathfrak {p}}\in \mathbb {N} _{0}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mathfrak {p}}>0}">

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<annotation encoding="application/x-tex">{\displaystyle e_{\mathfrak {p}}>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a967db854d1b15ec7b20631a6f5036bd182c3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.399ex; height:2.843ex;" alt="{\displaystyle e_{\mathfrak {p}}>0}" loading="lazy"> für nur endlich viele Primideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">. Für höchstens <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48272d5cc1b9ef13da57bed443a4aceac5850e4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.088ex; height:2.009ex;" alt="{\displaystyle n_{K}}" loading="lazy"> viele Primideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mathfrak {p}}>0}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle e_{\mathfrak {p}}>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a967db854d1b15ec7b20631a6f5036bd182c3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.399ex; height:2.843ex;" alt="{\displaystyle e_{\mathfrak {p}}>0}" loading="lazy"> gelten. Solche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> teilen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)}">

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<annotation encoding="application/x-tex">{\displaystyle (p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b43fe924fbe82623de1aa506862fc522c5c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle (p)}" loading="lazy"> und man schreibt dafür <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}|(p)}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}|(p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/185ea6e582f30d401e343d4cee5c5c924e07e239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.788ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}|(p)}" loading="lazy">. Der Exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/278e9c04f8313b3c5321479a1514d19e48da919f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.138ex; height:2.343ex;" alt="{\displaystyle e_{\mathfrak {p}}}" loading="lazy"> in der Primidealzerlegung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)}">

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<annotation encoding="application/x-tex">{\displaystyle (p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b43fe924fbe82623de1aa506862fc522c5c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle (p)}" loading="lazy"> heißt der Verzweigungsindex von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}|(p)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}|(p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/185ea6e582f30d401e343d4cee5c5c924e07e239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.788ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}|(p)}" loading="lazy">, so gilt

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}({\mathfrak {p}})=p^{f_{\mathfrak {p}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3cb668e2202ca0eeeb0f671a7633e030be5a350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:11.446ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}({\mathfrak {p}})=p^{f_{\mathfrak {p}}}}" loading="lazy">

für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathfrak {p}}\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle f_{\mathfrak {p}}\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b7d460a6c089166915594162b2b4095cdc1f22d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:2.843ex;" alt="{\displaystyle f_{\mathfrak {p}}\in \mathbb {N} }" loading="lazy">, welches der Trägheitsindex von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> genannt wird. Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy"> erfüllen die zum Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b43fe924fbe82623de1aa506862fc522c5c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle (p)}" loading="lazy"> gehörenden Verzweigungs- und Trägheitsindizes die folgende Beziehung zum Grad von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce6782a2935122dafec2947d1d7ba168be4399ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }" loading="lazy">:

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{{\mathfrak {p}}|(p)}e_{\mathfrak {p}}f_{\mathfrak {p}}=n_{K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49096affa2bd03289bd9ecc6c29f2a2749332e96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.937ex; height:6.009ex;" alt="{\displaystyle \sum _{{\mathfrak {p}}|(p)}e_{\mathfrak {p}}f_{\mathfrak {p}}=n_{K}.}" loading="lazy"></dd></dl>

Mit Hilfe der Kenntnis der Trägheitsindizes für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy"> lassen sich nun die lokalen Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}(K,p)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}(K,p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d73795bf270c7a0dced23c058626934a646527b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.476ex; height:3.009ex;" alt="{\displaystyle \alpha _{j}(K,p)}" loading="lazy"> bestimmen, nämlich über die Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-p^{-sf_{\mathfrak {p}}})^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle (1-p^{-sf_{\mathfrak {p}}})^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99aeab8435b8ab205037a5bbdf55d2c00a861662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.233ex; height:3.176ex;" alt="{\displaystyle (1-p^{-sf_{\mathfrak {p}}})^{-1}}" loading="lazy"> in der Identität<a href="#cite_note-29">[28]</a>

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<annotation encoding="application/x-tex">{\displaystyle \prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}=\prod _{p\in \mathbb {P} }\;\prod _{{\mathfrak {p}}|(p)}(1-p^{-sf_{\mathfrak {p}}})^{-1},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31a1e0d6b2dc93e89c18c9e1d09ad2f266dbf1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:43.98ex; height:6.843ex;" alt="{\displaystyle \prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-{\mathcal {N}}({\mathfrak {p}})^{-s}}}=\prod _{p\in \mathbb {P} }\;\prod _{{\mathfrak {p}}|(p)}(1-p^{-sf_{\mathfrak {p}}})^{-1},}" loading="lazy"></dd></dl>

indem man die Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{f_{\mathfrak {p}}}-1}">

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<annotation encoding="application/x-tex">{\displaystyle X^{f_{\mathfrak {p}}}-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f91c88cbd4adf5c3d59bde2e4cc0686eb2f440d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.869ex; height:2.843ex;" alt="{\displaystyle X^{f_{\mathfrak {p}}}-1}" loading="lazy"> im <a href="Polynomring" title="Polynomring">Polynomring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [X]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [X]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6c0e47242946827f33aad1803d66106d2f0307e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [X]}" loading="lazy"> faktorisiert.

Der Gamma-Faktor bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

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<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy"> ist<a href="#cite_note-30">[29]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (K,s)=\pi ^{-{\frac {s\cdot n_{K}}{2}}}\,\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e62f81d7437de442980e881ba4009087316625e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.807ex; height:6.176ex;" alt="{\displaystyle \gamma (K,s)=\pi ^{-{\frac {s\cdot n_{K}}{2}}}\,\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}.}" loading="lazy"></dd></dl>

Der Betrag der Diskriminante von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist der Konduktor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

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<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle q(K)=|d_{K}|.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958412f0a0a4ca139535e4a753d48c2c0b9950dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.886ex; height:2.843ex;" alt="{\displaystyle q(K)=|d_{K}|.}" loading="lazy"></dd></dl>

Damit ist die vollständige L-Funktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (K,s):=q(K)^{\frac {s}{2}}\,\gamma (K,s)\,L(K,s)=\left({\frac {|d_{K}|}{\pi ^{n_{K}}}}\right)^{\frac {s}{2}}\,\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}\,\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffa00e7cdab1c0326361ab81f59c4cea061f4b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:87.699ex; height:8.176ex;" alt="{\displaystyle \Lambda (K,s):=q(K)^{\frac {s}{2}}\,\gamma (K,s)\,L(K,s)=\left({\frac {|d_{K}|}{\pi ^{n_{K}}}}\right)^{\frac {s}{2}}\,\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}\,\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {1}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}" loading="lazy"></dd></dl>

Diese besitzt eine analytische Fortsetzung auf die komplexe Zahlenebene mit einfachen Polen bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}">

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<annotation encoding="application/x-tex">{\displaystyle s=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7903b8069a44c70f6f96511675bdd9a4ff200ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}">

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<annotation encoding="application/x-tex">{\displaystyle s=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac386d8f227fb823cede9b3e33d706cad3ed306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}" loading="lazy"> und den dortigen Residuen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {2^{r}hR}{w}}}">

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {2^{r}hR}{w}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4fd8787dc58af5356db95a9b6538996a0cb31e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.883ex; height:5.343ex;" alt="{\displaystyle -{\frac {2^{r}hR}{w}}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2^{r}hR}{w}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2^{r}hR}{w}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b43589e41f92783d51f0ac3ceeaf0e22cc15601.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.075ex; height:5.343ex;" alt="{\displaystyle {\frac {2^{r}hR}{w}}}" loading="lazy">. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=n_{1}+n_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r=n_{1}+n_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/084221780b35fa071c319cb6580c0aa67f8d73ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.885ex; height:2.343ex;" alt="{\displaystyle r=n_{1}+n_{2}}" loading="lazy"> die Anzahl der unendlichen Stellen, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd796ca77a508511c1ba760a6b508973b08afc36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.858ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {N} }" loading="lazy"> die <a href="Klassenzahl" title="Klassenzahl">Klassenzahl</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle R\in \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8efabc559c24b13880027bd80a44f2323696e021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.283ex; height:2.176ex;" alt="{\displaystyle R\in \mathbb {R} }" loading="lazy"> der <a href="Regulator_(Zahlentheorie)" title="Regulator (Zahlentheorie)">Regulator</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle w\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc1e4a9f74189220021305e0f413e06279dcf410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.183ex; height:2.176ex;" alt="{\displaystyle w\in \mathbb {N} }" loading="lazy"> die Anzahl der <a href="Einheitswurzel" title="Einheitswurzel">Einheitswurzeln</a>, die in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> liegen.<a href="#cite_note-32">[31]</a>

Dedekindsche L-Funktionen haben stets die Wurzelzahl 1:

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<annotation encoding="application/x-tex">{\displaystyle \epsilon (K)=1.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea9be7177080ace00a51f191ee37641f70289c58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.727ex; height:2.843ex;" alt="{\displaystyle \epsilon (K)=1.}" loading="lazy"></dd></dl>

Somit genügt die analytisch fortgesetzte, vollständige L-Funktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> der Funktionalgleichung<a href="#cite_note-34">[33]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (K,s)=\Lambda (K,1-s).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6785217eefa4551ea8cafd1766dc544fdfaf6697.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.974ex; height:2.843ex;" alt="{\displaystyle \Lambda (K,s)=\Lambda (K,1-s).}" loading="lazy"></dd></dl>

Die analytisch fortgesetzte Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (K,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (K,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/000865aed59c509f6b74393ecb4b110ebcbca131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.613ex; height:2.843ex;" alt="{\displaystyle \Lambda (K,s)}" loading="lazy"> gestattet nun auch die analytische Fortsetzung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

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<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy">, nämlich durch die Definition<a href="#cite_note-35">[34]</a>

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<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(K,s):={\frac {\Lambda (K,s)}{|d_{k}|^{\frac {s}{2}}\,\gamma (K,s)}}=\left({\frac {\pi ^{n_{K}}}{|d_{K}|}}\right)^{\frac {s}{2}}\cdot {\frac {\Lambda (K,s)}{\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73e0c90bf97dfdd0104e1538093b2dc7990d9812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:62.349ex; height:9.009ex;" alt="{\displaystyle L(K,s):={\frac {\Lambda (K,s)}{|d_{k}|^{\frac {s}{2}}\,\gamma (K,s)}}=\left({\frac {\pi ^{n_{K}}}{|d_{K}|}}\right)^{\frac {s}{2}}\cdot {\frac {\Lambda (K,s)}{\Gamma \left({\frac {s}{2}}\right)^{n_{1}+n_{2}}\,\Gamma \left({\frac {s+1}{2}}\right)^{n_{2}}}}.}" loading="lazy"></dd></dl>

Dadurch wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy"> zu einer meromorphen Funktion auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> mit einem einfachen Pol in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac386d8f227fb823cede9b3e33d706cad3ed306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}" loading="lazy">. Eine ihrer faszinierenden Eigenschaften ist die sogenannte <a href="Klassenzahlformel" title="Klassenzahlformel">analytische Klassenzahlformel</a>, wonach das Residuum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(K,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(K,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465aba2c976a2521c78ab53ebcf3070e3931c844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:2.843ex;" alt="{\displaystyle L(K,s)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac386d8f227fb823cede9b3e33d706cad3ed306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}" loading="lazy"> die folgende Gestalt annimmt:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mrow>

<msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Res} _{s=1}\,L(K,s)={\frac {2^{n_{1}}(2\pi )^{n_{2}}}{w{\sqrt {|d_{K}|}}}}\,hR.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/960e62ba888c5b09494e2e377596a9240314c1bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.758ex; height:7.009ex;" alt="{\displaystyle \operatorname {Res} _{s=1}\,L(K,s)={\frac {2^{n_{1}}(2\pi )^{n_{2}}}{w{\sqrt {|d_{K}|}}}}\,hR.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Heckesche_L-Funktionen">Heckesche L-Funktionen</h3></div>

Heckesche L-Funktionen sind gemeinsame Verallgemeinerungen der Dirichletschen und der Dedekindschen L-Funktionen. Sie beziehen sich also einerseits auf beliebige algebraische Zahlkörper (wie die Dedekindschen L-Funktionen) und hängen andererseits von geeigneten Charakteren ab (wie die Dirichletschen L-Funktionen). Der deutsche Mathematiker Erich Hecke definierte die nach ihm benannten L-Funktionen mit Hilfe sogenannter Größencharaktere und konnte die bei L-Funktionen gewünschten Eigenschaften beweisen. Der modernere Zugang zu L-Funktionen mit Bezug zu beliebigen algebraischen Zahlkörpern und geeigneten Charakteren, der auch noch weitreichend verallgemeinert werden kann, verwendet Idelklassencharaktere.

<div class="mw-heading mw-heading4"><h4 id="L-Funktionen_zu_Größencharakteren">L-Funktionen zu Größencharakteren</h4></div>

Heckesche L-Reihen zu Größencharakteren besitzen die Form<a href="#cite_note-37">[36]</a>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {\chi ({\mathfrak {a}})}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d957efdf60073ba51ae2e78ab981d9bb8b03cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.05ex; height:7.343ex;" alt="{\displaystyle L(\chi ,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {\chi ({\mathfrak {a}})}{{\mathcal {N}}({\mathfrak {a}})^{s}}}.}" loading="lazy"></dd></dl>

Wie bei Dedekindschen L-Funktionen bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> einen algebraischen Zahlkörper mit Ganzheitsring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> und Erweiterungsgrad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n:=n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9170ad643d0dfc07d703daeec146ea3c6a210282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.95ex; height:2.843ex;" alt="{\displaystyle n:=n_{K}=[K:\mathbb {Q} ]\in \mathbb {N} }" loading="lazy">. Die Summe durchläuft wieder alle vom <a href="Nullideal" class="mw-redirect" title="Nullideal">Nullideal</a> verschiedenen, ganzen Ideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa778b8e4af4159a88c06b9a8f7d07aee59cdc78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:9.827ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}({\mathfrak {a}})\in \mathbb {N} }" loading="lazy"> bezeichnet die Absolutnorm von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">. Die komplexen Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ({\mathfrak {a}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi ({\mathfrak {a}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/991d4603dd6bba666a7019922b622547e4950531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.427ex; height:2.843ex;" alt="{\displaystyle \chi ({\mathfrak {a}})}" loading="lazy"> beruhen auf einem Charakter (d. h. Gruppenhomomorphismus)

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}:=\{z\in \mathbb {C} :|z|=1\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ad9eef3de48eb92fa66d3e3233d0c6f991a0192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.922ex; height:3.176ex;" alt="{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}:=\{z\in \mathbb {C} :|z|=1\}.}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> ein ganzes Ideal von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mathfrak {m}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58d69c6cef0c0e54da6f0875a3ae287a53d4d4d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.019ex; height:2.343ex;" alt="{\displaystyle J^{\mathfrak {m}}}" loading="lazy"> symbolisiert die Gruppe der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilerfremden, <a href="Gebrochenes_Ideal" title="Gebrochenes Ideal">gebrochenen Ideale</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Das bedeutet: ein gebrochenes Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47431c0c547b479e2cb895f75e15aefdb16bfd80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> liegt genau dann in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mathfrak {m}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58d69c6cef0c0e54da6f0875a3ae287a53d4d4d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.019ex; height:2.343ex;" alt="{\displaystyle J^{\mathfrak {m}}}" loading="lazy">, wenn der Exponent von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> in der Primidealzerlegung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47431c0c547b479e2cb895f75e15aefdb16bfd80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}" loading="lazy"> gleich 0 ist für alle Primideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, die das ganze Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilen (d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\supseteq {\mathfrak {m}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\supseteq {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e104da0c92651edb2c613a37e74e4f44b69e0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.044ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}\supseteq {\mathfrak {m}}}" loading="lazy">). Die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{\mathfrak {m}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58d69c6cef0c0e54da6f0875a3ae287a53d4d4d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.019ex; height:2.343ex;" alt="{\displaystyle J^{\mathfrak {m}}}" loading="lazy"> verallgemeinert die bei Dirichletschen L-Funktionen verwendeten Gruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /m)^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /m)^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01c79e8a7a62f4426a18006bfc9e76b29ac59323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.073ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /m)^{\times }}" loading="lazy">.

Ist nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e240ba57afdb44beb8b205edc94e398047f894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.018ex; height:3.009ex;" alt="{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}" loading="lazy"> ein beliebiger, solcher Charakter und setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ({\mathfrak {a}})=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi ({\mathfrak {a}})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14eb64e4091be24ae69741f98f7446cda4a61feb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.688ex; height:2.843ex;" alt="{\displaystyle \chi ({\mathfrak {a}})=0}" loading="lazy"> für alle Ideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy">, die nicht teilerfremd zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> sind, so konvergiert die oben angegebene L-Reihe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df2ede56cb317a2561bcfa48a1ea433b274781bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle L(\chi ,s)}" loading="lazy"> in der Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> absolut. Wegen der Eindeutigkeit der Primidealzerlegung von Idealen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> gilt die nachfolgende Gleichheit mit dem zugehörenden Euler-Produkt, welches alle von Null verschiedenen Primideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> durchläuft:<a href="#cite_note-38">[37]</a>

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<msub>

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</munder>

<mfrac>

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<msup>

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</mfrac>

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<msub>

</mrow>

</mrow>

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</munder>

<mfrac>

<mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {\chi ({\mathfrak {a}})}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-\chi ({\mathfrak {p}}){\mathcal {N}}({\mathfrak {p}})^{-s}}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/735350138a7c9d3f41dca539ed6df2dcd3c15c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.189ex; height:7.343ex;" alt="{\displaystyle L(\chi ,s)=\sum _{0\neq {\mathfrak {a}}\subset {\mathcal {O}}_{K}}{\frac {\chi ({\mathfrak {a}})}{{\mathcal {N}}({\mathfrak {a}})^{s}}}=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\frac {1}{1-\chi ({\mathfrak {p}}){\mathcal {N}}({\mathfrak {p}})^{-s}}}.}" loading="lazy"></dd></dl>

Die eigentliche Herausforderung liegt nun aber in einer geeigneten Auswahl der Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74e240ba57afdb44beb8b205edc94e398047f894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.018ex; height:3.009ex;" alt="{\displaystyle \chi :\;J^{\mathfrak {m}}\;\longrightarrow \;S^{1}}" loading="lazy">, so dass die für L-Funktionen typischen Eigenschaften bewiesen werden können. Die Charaktere mit diesen wünschenswerten Eigenschaften heißen Größencharaktere: Ein Größencharakter modulo eines ganzen Ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> des algebraischen Zahlkörpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist ein Gruppenhomomorphismus<a href="#cite_note-39">[38]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :\;J^{\mathfrak {m}}\longrightarrow \;S^{1},\;\;{\mathfrak {b}}\longmapsto \chi ({\mathfrak {b}}),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a065661f25c7f2db876c4d2086eb8546f3494ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.09ex; height:3.176ex;" alt="{\displaystyle \chi :\;J^{\mathfrak {m}}\longrightarrow \;S^{1},\;\;{\mathfrak {b}}\longmapsto \chi ({\mathfrak {b}}),}" loading="lazy"></dd></dl>

zu dem es zwei Charaktere

<msub>

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<msub>

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</msub>

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<msup>

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</msup>

<msup>

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</msup>

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}:\;({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }\longrightarrow S^{1}\;{\text{ und }}\;\chi _{\infty }:\;\mathbf {R} ^{\star }\longrightarrow S^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f7d1c0357610a9e601ca2c701d3cfc734a38c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.03ex; height:3.176ex;" alt="{\displaystyle \chi _{e}:\;({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }\longrightarrow S^{1}\;{\text{ und }}\;\chi _{\infty }:\;\mathbf {R} ^{\star }\longrightarrow S^{1}}" loading="lazy"></dd></dl>

gibt, so dass für alle zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilerfremden Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70183d2be9b1ffc4d013cbf9f07ba160a46fd35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.614ex; height:2.509ex;" alt="{\displaystyle a\in {\mathcal {O}}_{K}}" loading="lazy"> gilt:

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \chi ((a))=\chi _{e}(a)\,\chi _{\infty }(a).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/004999e350c73618ec1c06be1c95f14bc40b3060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.298ex; height:2.843ex;" alt="{\displaystyle \chi ((a))=\chi _{e}(a)\,\chi _{\infty }(a).}" loading="lazy"></dd></dl>

Zur Erläuterung dieser Definition:

<msub>

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</msub>

</mrow>

</mrow>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f45b2f028f432322173ef983ecad253e3627dd78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.352ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }}" loading="lazy"> symbolisiert die <a href="Einheitengruppe" title="Einheitengruppe">Einheitengruppe</a> des <a href="Faktorring" title="Faktorring">Restklassenrings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">, besteht also aus allen Elementen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {m}}}">

<msub>

</mrow>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ad46f6b137cd110fb3dca40945f76528c9d6b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.488ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {m}}}" loading="lazy">, die invertierbar sind.</li></ul>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> bezeichnet den Minkowski-Raum bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> die Menge aller Einbettungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau :\;K\hookrightarrow \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau :\;K\hookrightarrow \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a334ae80288485681ac7be51d9c042dc1a8497b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.435ex; height:2.176ex;" alt="{\displaystyle \tau :\;K\hookrightarrow \mathbb {C} }" loading="lazy">, so besteht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> aus allen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-Tupeln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z_{\tau })_{\tau \in X}}">

<msub>

</mrow>

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<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (z_{\tau })_{\tau \in X}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9ad0a177e5d6b5c6f2110ff82bd4b2956f3dd26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.551ex; height:2.843ex;" alt="{\displaystyle (z_{\tau })_{\tau \in X}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{\tau }\in \mathbb {C} }">

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</msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{\tau }\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/946c1363d3dd13a1030bc65843c9692fb9bb3124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle z_{\tau }\in \mathbb {C} }" loading="lazy">, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{\overline {\tau }}={\overline {(z_{\tau })}}}">

<msub>

<mover>

</mover>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{\overline {\tau }}={\overline {(z_{\tau })}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e492f8cf713fac7634e17e152b6dac5a96860ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.528ex; height:3.676ex;" alt="{\displaystyle z_{\overline {\tau }}={\overline {(z_{\tau })}}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau \in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7c1e47bcb12d0cf7bfb5bb7686bf741c22b8fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.023ex; height:2.176ex;" alt="{\displaystyle \tau \in X}" loading="lazy">.<a href="#cite_note-40">[39]</a> Addition und Multiplikation in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-<a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> sind komponentenweise definiert. Das Bild der Einbettungsfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\ni a\mapsto (\tau (a))_{\tau \in X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\ni a\mapsto (\tau (a))_{\tau \in X}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/250410fca372eee4919aeeb25847080c5e39a610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:2.843ex;" alt="{\displaystyle K\ni a\mapsto (\tau (a))_{\tau \in X}}" loading="lazy"> liegt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy">.<a href="#cite_note-41">[40]</a> Die multiplikative Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{\star }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14bbffd7810e1bb8a7fff6a82ac4157b6244ad7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{\star }}" loading="lazy"> besteht aus allen Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy">, bei denen sämtliche Komponenten von Null verschieden sind. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K^{\star }:=K\setminus \{0\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in K^{\star }:=K\setminus \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ff443567d8d083d757b58e29359a1630ed14c10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.712ex; height:2.843ex;" alt="{\displaystyle a\in K^{\star }:=K\setminus \{0\}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (a)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau (a)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f159c20bbe7fd157fdec5aab7346cdbebfbf370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle \tau (a)\neq 0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau \in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7c1e47bcb12d0cf7bfb5bb7686bf741c22b8fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.023ex; height:2.176ex;" alt="{\displaystyle \tau \in X}" loading="lazy">, denn die Einbettungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau \in X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7c1e47bcb12d0cf7bfb5bb7686bf741c22b8fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.023ex; height:2.176ex;" alt="{\displaystyle \tau \in X}" loading="lazy"> sind <a href="K%C3%B6rperhomomorphismus" title="Körperhomomorphismus">Körperhomomorphismen</a>. Der Gruppenhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{\infty }:\;K^{\star }\rightarrow \mathbf {R} ^{\star },\;a\mapsto \iota _{\infty }(a):=(\tau (a))_{\tau \in X}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota _{\infty }:\;K^{\star }\rightarrow \mathbf {R} ^{\star },\;a\mapsto \iota _{\infty }(a):=(\tau (a))_{\tau \in X}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89b3242d3f07cd20f2ffa32a0582341b5da9da77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.736ex; height:2.843ex;" alt="{\displaystyle \iota _{\infty }:\;K^{\star }\rightarrow \mathbf {R} ^{\star },\;a\mapsto \iota _{\infty }(a):=(\tau (a))_{\tau \in X}}" loading="lazy">, bettet also die multiplikative Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16c1d5c4024a1af0c4d2280c62b6949d9947ca03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.148ex; height:2.343ex;" alt="{\displaystyle K^{\star }}" loading="lazy"> in die multiplikative Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{\star }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14bbffd7810e1bb8a7fff6a82ac4157b6244ad7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{\star }}" loading="lazy"> ein.</li></ul>

<ul><li>Ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K^{\star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in K^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c91fbf319b7bc8f08247a738ab73eda0b884c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.219ex; height:2.343ex;" alt="{\displaystyle a\in K^{\star }}" loading="lazy"> heißt teilerfremd zum ganzen Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">, wenn das Hauptideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a):=a{\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a):=a{\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fb28894076a7fc6bd51c3dc993b5bd730929338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.557ex; height:2.843ex;" alt="{\displaystyle (a):=a{\mathcal {O}}_{K}}" loading="lazy"> teilerfremd zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> ist. Bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{({\mathfrak {m}})}:=\{a\in K^{\star }\mid (a,{\mathfrak {m}})=1\}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{({\mathfrak {m}})}:=\{a\in K^{\star }\mid (a,{\mathfrak {m}})=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af923d0a8eff4c917353152cb48119156889af0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.209ex; height:3.343ex;" alt="{\displaystyle K^{({\mathfrak {m}})}:=\{a\in K^{\star }\mid (a,{\mathfrak {m}})=1\}}" loading="lazy"> die Gruppe der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilerfremden Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K^{\star }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in K^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c91fbf319b7bc8f08247a738ab73eda0b884c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.219ex; height:2.343ex;" alt="{\displaystyle a\in K^{\star }}" loading="lazy"> und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b/c}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=b/c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f37c80269fefb89251e1e003e2fdcbe179dd88c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.495ex; height:2.843ex;" alt="{\displaystyle a=b/c}" loading="lazy"> mit zwei zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilerfremden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,c\in {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b,c\in {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaece59e635daee9b948bb607248ccc66f8ac600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.422ex; height:2.509ex;" alt="{\displaystyle b,c\in {\mathcal {O}}_{K}}" loading="lazy">, so hat man den wohldefinierten Gruppenhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{e}:\;K^{({\mathfrak {m}})}\rightarrow ({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota _{e}:\;K^{({\mathfrak {m}})}\rightarrow ({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bb6e1412e9104cae7121113d885fff1ca721bbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.236ex; height:3.343ex;" alt="{\displaystyle \iota _{e}:\;K^{({\mathfrak {m}})}\rightarrow ({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto [b+{\mathfrak {m}}]/[c+{\mathfrak {m}}]}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\mapsto [b+{\mathfrak {m}}]/[c+{\mathfrak {m}}]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d0ec9ce31b5c867f86f880d217817332b2ef2ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.844ex; height:2.843ex;" alt="{\displaystyle a\mapsto [b+{\mathfrak {m}}]/[c+{\mathfrak {m}}]}" loading="lazy">, der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b/c}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=b/c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f37c80269fefb89251e1e003e2fdcbe179dd88c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.495ex; height:2.843ex;" alt="{\displaystyle a=b/c}" loading="lazy"> auf den Quotienten der Restklassen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> abbildet.<a href="#cite_note-42">[41]</a></li></ul>

<ul><li>Entsprechend seiner Definition zerfällt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> auf den zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> teilerfremden Hauptidealen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a)\subset {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a)\subset {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/703e08b046b14b16546d452efb1b5effb5cc41e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.681ex; height:2.843ex;" alt="{\displaystyle (a)\subset {\mathcal {O}}_{K}}" loading="lazy"> in einen „endlichen“ Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> und einen „unendlichen“ Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84f033618aafdfef6f1031e8ffcc1669cd09a500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.009ex;" alt="{\displaystyle \chi _{\infty }}" loading="lazy">. Korrekterweise müssten in dieser Zerlegungsbedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06b3a067fa20a0a0cfe928d9295ca686366b810f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.493ex; height:2.843ex;" alt="{\displaystyle \chi _{e}(a)}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}(\iota _{e}(a))}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}(\iota _{e}(a))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37a265a5e24d04c9934f5f9fb7944cf51d74f8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.124ex; height:2.843ex;" alt="{\displaystyle \chi _{e}(\iota _{e}(a))}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\infty }(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\infty }(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62902331f2b378800b7adf975b2a73d8d73ba310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.37ex; height:2.843ex;" alt="{\displaystyle \chi _{\infty }(a)}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\infty }(\iota _{\infty }(a))}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\infty }(\iota _{\infty }(a))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb5ebbe3bca42f013a12ac72fb69876efacd5c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.878ex; height:2.843ex;" alt="{\displaystyle \chi _{\infty }(\iota _{\infty }(a))}" loading="lazy"> ersetzt werden, worauf der Kürze halber stets verzichtet wird. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84f033618aafdfef6f1031e8ffcc1669cd09a500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.009ex;" alt="{\displaystyle \chi _{\infty }}" loading="lazy"> sind durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> eindeutig bestimmt.<a href="#cite_note-43">[42]</a></li></ul>

Ein Größencharakter modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> heißt primitiv, wenn er nicht als Einschränkung

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<annotation encoding="application/x-tex">{\displaystyle \chi =\chi '|_{J^{m}}:J^{\mathfrak {m}}\longrightarrow S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceabaf832869d1d5ae9345dcf4b66a7b2f6db4f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.615ex; height:3.343ex;" alt="{\displaystyle \chi =\chi '|_{J^{m}}:J^{\mathfrak {m}}\longrightarrow S^{1}}" loading="lazy"></dd></dl>

eines Größencharakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ':\,J^{{\mathfrak {m}}'}\rightarrow S^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ':\,J^{{\mathfrak {m}}'}\rightarrow S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4bf54e50a6f82731cb0e4553b727de7f7b63126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.204ex; height:3.176ex;" alt="{\displaystyle \chi ':\,J^{{\mathfrak {m}}'}\rightarrow S^{1}}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}'}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab321a35cf4bb324d5e9e90c465481ab5cce457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {m}}'}" loading="lazy"> dargestellt werden kann, in der das ganze Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}'}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab321a35cf4bb324d5e9e90c465481ab5cce457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {m}}'}" loading="lazy"> ein echter Teiler des ganzen Ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> ist. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> ist genau dann nicht primitiv, wenn sein „endlicher“ Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {m}}')^{\star }}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {m}}')^{\star }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0003745d45d787b5dbf8fa20b0f8152494c612c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.036ex; height:3.009ex;" alt="{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {m}}')^{\star }}" loading="lazy"> faktorisiert, d. h. wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> als Kompositum

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}:\;({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }\longrightarrow ({\mathcal {O}}_{K}/{\mathfrak {m}}')^{\star }\;{\stackrel {\chi _{e}'}{\longrightarrow }}\;S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d512ac46dc9020d1bd26dbbe7677a2e030b31d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.191ex; height:4.343ex;" alt="{\displaystyle \chi _{e}:\;({\mathcal {O}}_{K}/{\mathfrak {m}})^{\star }\longrightarrow ({\mathcal {O}}_{K}/{\mathfrak {m}}')^{\star }\;{\stackrel {\chi _{e}'}{\longrightarrow }}\;S^{1}}" loading="lazy"></dd></dl>

geschrieben werden kann, in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}'}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{e}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33d7a02a3b8a8601d0a51a489f2ec9ab2c0fa621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.509ex;" alt="{\displaystyle \chi _{e}'}" loading="lazy"> der „endliche“ Charakter eines Größencharakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi '}">

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<annotation encoding="application/x-tex">{\displaystyle \chi '}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b91c8ab9abc0c598ff97862779af8dfe87d9a347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.14ex; height:2.843ex;" alt="{\displaystyle \chi '}" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}'}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab321a35cf4bb324d5e9e90c465481ab5cce457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {m}}'}" loading="lazy"> ist mit einem echten Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}'}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fab321a35cf4bb324d5e9e90c465481ab5cce457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {m}}'}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">. Der Führer eines Größencharakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> ist der kleinste Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d7c8d65d0a0dbdbcd9c31e1791351016f30a445.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.758ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {f}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> als Einschränkung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi '|_{J^{\mathfrak {m}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi '|_{J^{\mathfrak {m}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf67b0fbfeddcc3c456dd278258f41ef2cc99c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.285ex; height:3.176ex;" alt="{\displaystyle \chi '|_{J^{\mathfrak {m}}}}" loading="lazy"> eines Größencharakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ':J^{\mathfrak {f}}\rightarrow S^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ':J^{\mathfrak {f}}\rightarrow S^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4eae42c31c51da244e114f7860f99297017d6ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.561ex; height:3.009ex;" alt="{\displaystyle \chi ':J^{\mathfrak {f}}\rightarrow S^{1}}" loading="lazy"> dargestellt werden kann. Der Führer von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> lässt sich auch mit Hilfe des „endlichen“ Charakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> definieren: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d7c8d65d0a0dbdbcd9c31e1791351016f30a445.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.758ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {f}}}" loading="lazy"> ist der kleinste Teiler von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6fe923dba59b5e55647010177706de2aa5e8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.454ex; height:2.009ex;" alt="{\displaystyle \chi _{e}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {f}})^{\star }}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {f}})^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47233fe1791fd49908dc8fc93c5ca0bc0bc41c47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.327ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {O}}_{K}/{\mathfrak {f}})^{\star }}" loading="lazy"> faktorisiert.<a href="#cite_note-44">[43]</a>

Mit Hilfe des Begriffs eines primitiven Größencharakters lassen sich nun alle für L-Funktionen typischen Objekte definieren und die gewünschten Aussagen nachweisen: vervollständigte Heckesche L-Funktion, analytische Fortsetzung, Führer, Wurzelzahl und Funktionalgleichung. Dies ist der Weg, den Erich Hecke beschritten hat und im Lehrbuch von Neukirch detailliert beschrieben wird.<a href="#cite_note-45">[44]</a> Dieser Zugang verwendet allerdings mathematische Objekte (ideale Zahlen), die aus heutiger Sicht als überholt gelten können und das eigentliche Wesen der vervollständigten Heckeschen L-Funktionen verschleiern.<a href="#cite_note-46">[45]</a>

<div class="mw-heading mw-heading4"><h4 id="L-Funktionen_zu_Idelklassencharakteren">L-Funktionen zu Idelklassencharakteren</h4></div>

Die modernere Theorie zur Behandlung von Heckeschen L-Funktionen, die auch noch weitreichend verallgemeinert werden kann, ist unter dem Namen Tate's Thesis (Doktorarbeit des US-amerikanischen Mathematikers <a href="John_T._Tate" title="John T. Tate">John T. Tate</a>) bekannt. Diese Theorie verwendet Idelklassencharaktere anstelle primitiver Größencharaktere<a href="#cite_note-47">[46]</a>, zu deren Definition die Begriffe Adel und Idel benötigt werden: Sei weiterhin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein algebraischer Zahlkörper und

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</munder>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} :=\mathbb {A} _{K}:=\prod _{v}'K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd7158684e268bcd8254dfa1d45731eebd0769c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.584ex; height:5.676ex;" alt="{\displaystyle \mathbb {A} :=\mathbb {A} _{K}:=\prod _{v}'K_{v}}" loading="lazy"></dd></dl>

der <a href="Adelring" class="mw-redirect" title="Adelring">Adelring</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Dabei durchläuft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> die <a href="Unendliche_Menge" title="Unendliche Menge">unendliche Menge</a> aller Stellen, d. h. die Menge aller Äquivalenzklassen nicht-trivialer <a href="Betragsfunktion" title="Betragsfunktion">Absolutbeträge</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> bezeichnet die komplette Hülle (Vervollständigung) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">.<a href="#cite_note-48">[Anm. 2]</a> Das Auslassungszeichen am Produktsymbol zeigt die Einschränkung im Vergleich zum direkten Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{v}K_{v}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{v}K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc7cdcc590c36999ddc2567dd43a62308aeea180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.359ex; height:5.509ex;" alt="{\displaystyle \prod _{v}K_{v}}" loading="lazy"> an, dass in jedem Adel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{v})_{v}\in \mathbb {A} }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{v})_{v}\in \mathbb {A} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a20aafae1ccd2dee7e0f18350af1b51b0147405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.717ex; height:2.843ex;" alt="{\displaystyle (x_{v})_{v}\in \mathbb {A} }" loading="lazy"> für fast alle (d. h. für alle bis auf höchstens endlich viele) Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> die Komponente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}\in K_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{v}\in K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d11c059ef4b97bb0c0673d672b11df55ad5e58a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.203ex; height:2.509ex;" alt="{\displaystyle x_{v}\in K_{v}}" loading="lazy"> im Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4714e4b7795af9e2c3d727be09bd091a4d1076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{v}}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">-ganzen Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> liegen muss. Der kanonische Einbettungshomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\hookrightarrow \mathbb {A} ,\;x\mapsto (x_{v})_{v}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\hookrightarrow \mathbb {A} ,\;x\mapsto (x_{v})_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/032219b35db8fd78d2702533de6baedadd02be48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.472ex; height:2.843ex;" alt="{\displaystyle K\hookrightarrow \mathbb {A} ,\;x\mapsto (x_{v})_{v}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}=x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{v}=x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/719f316c5b1a922ac0fabf0a6db016d82281dd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.787ex; height:2.009ex;" alt="{\displaystyle x_{v}=x}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, ist <a href="Wohldefiniertheit" title="Wohldefiniertheit">wohldefiniert</a>, da jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9075ef2df706bf2764f76182a1ddf55ff231a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}" loading="lazy"> für fast alle Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> ganz ist.<a href="#cite_note-49">[47]</a>

Die Elemente der multiplikativen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{\times }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cf472795ae7becafbdaed5e640887c03b3a2f41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {A} ^{\times }}" loading="lazy"> der <a href="Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheiten</a> (invertierbaren Elemente) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fb423c16a5f403edbaf66438b75e7a36e725af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }" loading="lazy"> heißen die <a href="Idelgruppe" class="mw-redirect" title="Idelgruppe">Idele</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{\times }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cf472795ae7becafbdaed5e640887c03b3a2f41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {A} ^{\times }}" loading="lazy"> lässt sich als eingeschränktes Produkt schreiben:

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</munder>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }=\prod _{v}'K_{v}^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9fc5936868c824b73dbf93e47733ddb863420d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.933ex; height:5.676ex;" alt="{\displaystyle \mathbb {A} ^{\times }=\prod _{v}'K_{v}^{\times }}" loading="lazy">.</dd></dl>

Hier bedeutet das Auslassungszeichen, dass ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{v})_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{v})_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b4dc527016969d4b54f8ebf17a6ca57e218ab10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.198ex; height:2.843ex;" alt="{\displaystyle (x_{v})_{v}}" loading="lazy"> des direkten Produkts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{v}K_{v}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{v}K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc7cdcc590c36999ddc2567dd43a62308aeea180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.359ex; height:5.509ex;" alt="{\displaystyle \prod _{v}K_{v}}" loading="lazy"> genau dann ein Idel ist, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e348e88c1cab743c6ec11cc9d7614d0367763a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}" loading="lazy"> für fast alle Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> gilt. Vermöge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}">

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f89e173c1dcc940492a5af01a8a3c42dca9f8d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.7ex; height:2.343ex;" alt="{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto (x_{v})_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto (x_{v})_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9cc3857c89bf161eba57e599d54d6f3d3cfd16e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.142ex; height:2.843ex;" alt="{\displaystyle x\mapsto (x_{v})_{v}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}=x}">

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<annotation encoding="application/x-tex">{\displaystyle x_{v}=x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/719f316c5b1a922ac0fabf0a6db016d82281dd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.787ex; height:2.009ex;" alt="{\displaystyle x_{v}=x}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, lässt sich die Einheitengruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> in die Gruppe der Idele einbetten, so dass man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98403ab9e2f27fde4576ca3e622add252f4fd9a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}" loading="lazy"> als Untergruppe der Idelgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{\times }}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cf472795ae7becafbdaed5e640887c03b3a2f41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {A} ^{\times }}" loading="lazy"> auffassen kann, deren Elemente die Hauptidele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> genannt werden. Die <a href="Faktorgruppe" title="Faktorgruppe">Quotientengruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{\times }/K^{\times }}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }/K^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b48721a7a45a3464b83edf6b23ae71550a78404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.956ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} ^{\times }/K^{\times }}" loading="lazy"> heißt die <a href="Idelklassengruppe" class="mw-redirect" title="Idelklassengruppe">Idelklassengruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Idelgruppe und Idelklassengruppe werden mit geeigneten <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologien</a> versehen, so dass man anschließend von stetigen Abbildungen sprechen kann.<a href="#cite_note-50">[48]</a> Ein Idelklassencharakter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist ein stetiger Gruppenhomomorphismus

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \chi :\mathbb {A} ^{\times }\longrightarrow \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b956f65b326e0668e61a14173dfbf6f558fc51b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.866ex; height:2.676ex;" alt="{\displaystyle \chi :\mathbb {A} ^{\times }\longrightarrow \mathbb {C} ^{\times }}" loading="lazy">,</dd></dl>

der auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}">

<msup>

</mrow>

</msup>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f89e173c1dcc940492a5af01a8a3c42dca9f8d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.7ex; height:2.343ex;" alt="{\displaystyle K^{\times }\hookrightarrow \mathbb {A} ^{\times }}" loading="lazy"> trivial ist, also dort ausschließlich den Wert 1 annimmt. Insofern kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> auch als Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{\times }/K^{\times }\rightarrow \mathbb {C} ^{\times }}">

<msup>

</mrow>

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<msup>

</mrow>

</msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{\times }/K^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8063b6e89e2f69915fe79aac1a1b3eb56c5d01e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.759ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} ^{\times }/K^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> der Idelklassengruppe auffassen, was den Namen „Idelklassencharakter“ rechtfertigt. Ein solcher Charakter besitzt stets eine Zerlegung

<msup>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \chi =\prod _{v}'\chi _{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/916148064c25daf9a6da0a421fe706c88f390388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.08ex; height:5.676ex;" alt="{\displaystyle \chi =\prod _{v}'\chi _{v}}" loading="lazy"></dd></dl>

in lokale Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0deef81ee472026f6cf789345c6d495c64d62e52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.83ex; height:2.676ex;" alt="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy">, die für fast alle Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> unverzweigt sind. Dies wird durch das Auslassungszeichen am Produktsymbol angezeigt. Dabei heißt ein lokaler Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32faa75b318d420fcea4f44307a1a25b3ebd628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}" loading="lazy"> unverzweigt, wenn seine Einschränkung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}^{\times }}">

</mrow>

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61ed696ad7cc8fcb8d5a657437ef72c3561b04b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.361ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{v}^{\times }}" loading="lazy"> trivial ist, andernfalls verzweigt.<a href="#cite_note-51">[49]</a>

Der Führer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}(\chi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}(\chi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0c408cf706a3b24557a8a772a93e9555b02c8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.023ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {f}}(\chi )}" loading="lazy"> eines Idelklassencharakters ist ein ganzes Ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy">, nämlich

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}(\chi )=\prod _{v\not \mid \infty }{\mathfrak {p}}_{v}^{a(\chi _{v})}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11d8895b272a155142b4ac5cd07f9bc2b0659d8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.509ex; height:6.176ex;" alt="{\displaystyle {\mathfrak {f}}(\chi )=\prod _{v\not \mid \infty }{\mathfrak {p}}_{v}^{a(\chi _{v})}.}" loading="lazy"></dd></dl>

Das Produkt durchläuft alle endlichen Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{v}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d71071c8b7002a2fde1d491a40bb0c24a3a2ba8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.192ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{v}}" loading="lazy"> ist das durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> bestimmte Primideal. Die Exponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\chi _{v})}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a(\chi _{v})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8614d11974d663985eb7ae6955fc85c962c29cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:2.843ex;" alt="{\displaystyle a(\chi _{v})}" loading="lazy"> liegen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{0}}">

<msub>

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</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77ab7e98123f0def29a1cd3df96a0b7a58f4202c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32faa75b318d420fcea4f44307a1a25b3ebd628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}" loading="lazy"> unverzweigt, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\chi _{v})=0}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(\chi _{v})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d03ca5b5279e30c2b5e9201de5458527c64e8a05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.785ex; height:2.843ex;" alt="{\displaystyle a(\chi _{v})=0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{v}^{a(\chi _{v})}}">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{v}^{a(\chi _{v})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40873771bb99843002e8bd4062ddee5ab9087d55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.384ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {p}}_{v}^{a(\chi _{v})}}" loading="lazy"> trägt nichts zum Führer bei. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32faa75b318d420fcea4f44307a1a25b3ebd628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}" loading="lazy"> verzweigt, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\chi _{v})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(\chi _{v})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8614d11974d663985eb7ae6955fc85c962c29cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:2.843ex;" alt="{\displaystyle a(\chi _{v})}" loading="lazy"> die kleinste natürliche Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d30d7dcf305b7bce39d36df72fe3985b47aa9961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32faa75b318d420fcea4f44307a1a25b3ebd628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\pi _{v}^{k}{\mathcal {O}}_{v}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+\pi _{v}^{k}{\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a0985545dbd45627496df9e472030ad8ea0df2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.305ex; height:2.843ex;" alt="{\displaystyle 1+\pi _{v}^{k}{\mathcal {O}}_{v}}" loading="lazy"> trivial ist. Dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{v}\in {\mathcal {O}}_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{v}\in {\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1a184322120ec3847327218854d11f65687719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.075ex; height:2.509ex;" alt="{\displaystyle \pi _{v}\in {\mathcal {O}}_{v}}" loading="lazy"> ein uniformisierendes Element.<a href="#cite_note-52">[50]</a>

Die (nicht vervollständigte) L-Funktion zum Idelklassencharakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> ist definiert durch

<msup>

</mrow>

</munder>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s):=\prod _{v\not \mid \infty }'(1-\chi _{v}(\pi _{v})q_{v}^{-s})^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e7f3cdd8af5f33e825b71a4b8870f2c40ed879.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:32.526ex; height:6.176ex;" alt="{\displaystyle L(\chi ,s):=\prod _{v\not \mid \infty }'(1-\chi _{v}(\pi _{v})q_{v}^{-s})^{-1}}" loading="lazy">.</dd></dl>

Das Produkt wird über alle endlichen Stellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> gebildet, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> unverzweigt ist. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f3050477d3efa6dc855debe33889d811c2a1a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.067ex; height:2.009ex;" alt="{\displaystyle q_{v}}" loading="lazy"> bezeichnet die Ordnung des <a href="Restklassenk%C3%B6rper" title="Restklassenkörper">Restklassenkörpers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}/(\pi _{v})}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}/(\pi _{v})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5df1be6c8d248dc84cade7481cbecf8bd2df6af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.206ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{v}/(\pi _{v})}" loading="lazy">. Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32faa75b318d420fcea4f44307a1a25b3ebd628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle \chi _{v}}" loading="lazy"> unverzweigt ist, hängt der Wert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}(\pi _{v})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}(\pi _{v})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9a122a4afb9f0faf381af77111a067d6cc20be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.649ex; height:2.843ex;" alt="{\displaystyle \chi _{v}(\pi _{v})}" loading="lazy"> nicht von der Wahl des uniformisierenden Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f21165a2b8dd291d122fe5b30a9625e978f671e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.355ex; height:2.009ex;" alt="{\displaystyle \pi _{v}}" loading="lazy"> ab. Das Produkt auf der rechten Seite der Definition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df2ede56cb317a2561bcfa48a1ea433b274781bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle L(\chi ,s)}" loading="lazy"> konvergiert in einer rechten Halbebene der komplexen Zahlenebene absolut.<a href="#cite_note-53">[51]</a>

Zur Vervollständigung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df2ede56cb317a2561bcfa48a1ea433b274781bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle L(\chi ,s)}" loading="lazy"> werden noch L-Faktoren an den unendlichen Stellen benötigt:<a href="#cite_note-54">[52]</a>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\infty }(\chi ,s):=\prod _{v\mid \infty }L(\chi _{v},s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535fb70f917cdc14435eab7c3d9014633cba6992.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.95ex; height:6.009ex;" alt="{\displaystyle L_{\infty }(\chi ,s):=\prod _{v\mid \infty }L(\chi _{v},s)}" loading="lazy">.</dd></dl>

Ist dabei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> eine reelle, unendliche Stelle, so ist die Vervollständigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. Der lokale Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0deef81ee472026f6cf789345c6d495c64d62e52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.83ex; height:2.676ex;" alt="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> kann also mit einem Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\times }\rightarrow \mathbb {C} ^{\times }}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd20c5cc866847bb08e287b83aba49e90782a821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.992ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> identifiziert werden. Letzterer besitzt notwendig die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto |t|^{s_{0}}(t/|t|)^{m}}">

</mrow>

<msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto |t|^{s_{0}}(t/|t|)^{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4656257b0b758d161df3e96d3b21168b3cef0c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.042ex; height:3.009ex;" alt="{\displaystyle t\mapsto |t|^{s_{0}}(t/|t|)^{m}}" loading="lazy"> mit eindeutig bestimmten Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}\in \mathbb {C} }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{0}\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2b4f4e2d29fad0140ad3312dbca64b6bc7f0f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.663ex; height:2.509ex;" alt="{\displaystyle s_{0}\in \mathbb {C} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \{0,1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in \{0,1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f95872d064ce94b55ab3a0a396f507c8699a741e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.565ex; height:2.843ex;" alt="{\displaystyle m\in \{0,1\}}" loading="lazy">. Der L-Faktor an der reellen, unendlichen Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> ist dann definiert durch

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi _{v},s):=\Gamma _{\mathbb {R} }(s+s_{0}+m),\;\Gamma _{\mathbb {R} }(z):=\pi ^{-z/2}\Gamma (z/2).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37d7679c8d73eec76e42fcb4fa8cf70e59080a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.158ex; height:3.343ex;" alt="{\displaystyle L(\chi _{v},s):=\Gamma _{\mathbb {R} }(s+s_{0}+m),\;\Gamma _{\mathbb {R} }(z):=\pi ^{-z/2}\Gamma (z/2).}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> eine nicht-reelle, unendliche, also komplexe Stelle, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> isomorph zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">, und es gibt zwei Möglichkeiten der Identifikation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">. Man wählt eine davon und kann dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0deef81ee472026f6cf789345c6d495c64d62e52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.83ex; height:2.676ex;" alt="{\displaystyle \chi _{v}:K_{v}^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> als Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{\times }\rightarrow \mathbb {C} ^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/299aeec87cf52da35ad7b5d58097e51b2530e373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.992ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> auffassen. Ein solcher Charakter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdb696059a558aeee27138277c41d30fd9bd7a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{\times }}" loading="lazy"> hat stets die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto |t|^{2s_{0}}(t/|t|)^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle t\mapsto |t|^{2s_{0}}(t/|t|)^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a722f8dfe539f5a7cf35ce756fe58a5f4f3f0196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.864ex; height:3.343ex;" alt="{\displaystyle t\mapsto |t|^{2s_{0}}(t/|t|)^{m}}" loading="lazy"> mit eindeutig bestimmten Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle s_{0}\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2b4f4e2d29fad0140ad3312dbca64b6bc7f0f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.663ex; height:2.509ex;" alt="{\displaystyle s_{0}\in \mathbb {C} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e01e20e2bc24ab21cca6a82edddec5c1a20ec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.431ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {Z} }" loading="lazy">. Der zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> gehörende L-Faktor ist dann gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle L(\chi _{v},s):=\Gamma _{\mathbb {C} }(s+s_{0}+|m|/2),\;\Gamma _{\mathbb {C} }(z):=2\cdot (2\pi )^{-z}\Gamma (z).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf71056acea10cc3a7612c4439ddfb5298fb0f0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.618ex; height:3.009ex;" alt="{\displaystyle L(\chi _{v},s):=\Gamma _{\mathbb {C} }(s+s_{0}+|m|/2),\;\Gamma _{\mathbb {C} }(z):=2\cdot (2\pi )^{-z}\Gamma (z).}" loading="lazy"><a href="#cite_note-55">[53]</a></dd></dl>

Die vollständige L-Funktion zum Idelklassencharakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> ist nun definiert als<a href="#cite_note-56">[54]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s):={\big (}|d_{K}|\cdot {\mathcal {N}}{\mathfrak {f(\chi )}}{\big )}^{s/2}\cdot L_{\infty }(\chi ,s)\cdot L(\chi ,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea8c00306ba32234775afefc33e9c255394ce3ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.874ex; height:3.843ex;" alt="{\displaystyle \Lambda (\chi ,s):={\big (}|d_{K}|\cdot {\mathcal {N}}{\mathfrak {f(\chi )}}{\big )}^{s/2}\cdot L_{\infty }(\chi ,s)\cdot L(\chi ,s)}" loading="lazy">,</dd></dl>

mit dem Absolutbetrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |d_{K}|}">

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<annotation encoding="application/x-tex">{\displaystyle |d_{K}|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b37a9d58f80a053666a06930962d24fa891c1ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.196ex; height:2.843ex;" alt="{\displaystyle |d_{K}|}" loading="lazy"> der Diskriminante von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und der Norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}{\mathfrak {f(\chi )}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}{\mathfrak {f(\chi )}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14b1277e8d95d438e1ab53814d1736fce6d9a758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:6.359ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}{\mathfrak {f(\chi )}}}" loading="lazy"> des Führers von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy">. Der Führer der L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> setzt sich also zusammen aus dem Absolutbetrag der Diskriminante des Zahlkörpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und der Norm des Führers des Idelklassencharakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy">:<a href="#cite_note-57">[55]</a>

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<annotation encoding="application/x-tex">{\displaystyle q(\chi ):=|d_{K}|\cdot {\mathcal {N}}{\mathfrak {f(\chi )}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab59c5e877e2ee2173da8ec420ac35649511c67b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.252ex; height:3.009ex;" alt="{\displaystyle q(\chi ):=|d_{K}|\cdot {\mathcal {N}}{\mathfrak {f(\chi )}}}" loading="lazy">.</dd></dl>

Der Grad von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> ist der Grad der Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce6782a2935122dafec2947d1d7ba168be4399ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }" loading="lazy">:<a href="#cite_note-58">[56]</a>

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<annotation encoding="application/x-tex">{\displaystyle d:=[K:\mathbb {Q} ]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c031592c8b6422be8db0a97ad189d922036fa556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.066ex; height:2.843ex;" alt="{\displaystyle d:=[K:\mathbb {Q} ]}" loading="lazy">.</dd></dl>

Was die Funktionalgleichung der vollständigen L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> angeht, so gibt es eine relle Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> und eine Wurzelzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (\chi )\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon (\chi )\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab01daeb1a976893b8bbba10fff7816dbca5fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.727ex; height:2.843ex;" alt="{\displaystyle \epsilon (\chi )\in \mathbb {C} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\epsilon (\chi )|=1}">

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<annotation encoding="application/x-tex">{\displaystyle |\epsilon (\chi )|=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43bb87bc9e817275802b46d757acd5a15e554b7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.763ex; height:2.843ex;" alt="{\displaystyle |\epsilon (\chi )|=1}" loading="lazy">, so dass gilt:<a href="#cite_note-59">[57]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)=\epsilon (\chi )\Lambda ({\overline {\chi }},r-s).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4133289c6b005468b6952976c052a6ac52c1eecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.962ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)=\epsilon (\chi )\Lambda ({\overline {\chi }},r-s).}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :\mathbb {A} ^{\times }\rightarrow \mathbb {C} ^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle \chi :\mathbb {A} ^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a27da5c728cc8809774a955c5e0c8259eb47998e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.384ex; height:2.676ex;" alt="{\displaystyle \chi :\mathbb {A} ^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> ein unitärer Idelklassencharakter, liegt also sein Bild auf dem Einheitskreis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d033f5052427c680d6de7e2f5f3266de96beab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.572ex; height:2.676ex;" alt="{\displaystyle {\mathcal {S}}^{1}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}">

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<annotation encoding="application/x-tex">{\displaystyle r=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e6584ba3b7843583b757896c2f0686efc0489e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}" loading="lazy">.<a href="#cite_note-60">[58]</a>

<div class="mw-heading mw-heading3"><h3 id="Artinsche_L-Funktionen">Artinsche L-Funktionen</h3></div>

Die bislang aufgeführten Beispiele von L-Funktionen beziehen sich direkt auf einzelne algebraische Zahlkörper und zugehörende Charaktere. Dem gegenüber sind Artinsche L-Funktionen den <a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungen</a> der <a href="Galoisgruppe" title="Galoisgruppe">Galoisgruppe</a> einer galoisschen Körpererweiterung zugeordnet. Sie stehen in einer engen Beziehung zu den oben genannten L-Funktionen, verallgemeinern jene jedoch erheblich. Die initiale Beobachtung, welche zur Definition von Artinschen L-Funktionen motiviert, betrachtet den Isomorphismus

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\;{\stackrel {\sim }{\longrightarrow }}\;G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f84a1c0f7d80ebe675807d14fec0fb05d8d013bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.02ex; height:3.343ex;" alt="{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\;{\stackrel {\sim }{\longrightarrow }}\;G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )}" loading="lazy"></dd></dl>

zwischen der Gruppe der invertierbaren Restklassen des Restklassenrings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /m\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /m\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fef120a86ce66f277c2cebb01f9c4c14de784de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.304ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /m\mathbb {Z} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42411e85d874a733209223302bbd8d5e3ad04cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {N} }" loading="lazy">, und der Galoisgruppe des <a href="Kreisteilungsk%C3%B6rper" title="Kreisteilungskörper">Kreisteilungskörpers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (\mu _{m})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\mu _{m})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d4f7a6f00be31f4e60273fec9c1740fe80873bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.694ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\mu _{m})}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-ten Einheitswurzeln. Dieser Gruppenisomorphismus ordnet der Restklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\operatorname {mod} m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\operatorname {mod} m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb444b2fed8819917b044aea76f3d7447445d661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.464ex; height:2.509ex;" alt="{\displaystyle p\operatorname {mod} m}" loading="lazy"> einer Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\not \mid m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\not \mid m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ec6a161df0abeeef2afd0edf8eddd308ecec149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle p\not \mid m}" loading="lazy"> den <a href="Frobeniusautomorphismus" class="mw-redirect" title="Frobeniusautomorphismus">Frobeniusautomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{p}:\mathbb {Q} (\mu _{m})/\mathbb {Q} \rightarrow \mathbb {Q} (\mu _{m})/\mathbb {Q} }">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{p}:\mathbb {Q} (\mu _{m})/\mathbb {Q} \rightarrow \mathbb {Q} (\mu _{m})/\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e15385aec2a5133f619aff2f7d17018a867cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.325ex; height:3.009ex;" alt="{\displaystyle \phi _{p}:\mathbb {Q} (\mu _{m})/\mathbb {Q} \rightarrow \mathbb {Q} (\mu _{m})/\mathbb {Q} }" loading="lazy"> zu, der jede <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-te Einheitswurzel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta ^{p}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta ^{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45081a8422870a947438520327d7450d5c41a62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.177ex; height:2.676ex;" alt="{\displaystyle \zeta ^{p}}" loading="lazy"> abbildet und dadurch festgelegt ist. Mit Hilfe dieser Isomorphie lässt sich ein Dirichlet-Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :(\mathbb {Z} /m\mathbb {Z} )^{\times }\rightarrow \mathbb {C} ^{\times }}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :(\mathbb {Z} /m\mathbb {Z} )^{\times }\rightarrow \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb1395e87c4227a502391ae210912dc79c5e6088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.819ex; height:2.843ex;" alt="{\displaystyle \chi :(\mathbb {Z} /m\mathbb {Z} )^{\times }\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> auch als ein Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi :G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )\rightarrow \mathbb {C} ^{\times }}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )\rightarrow \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f06d6c7e9d061214b0a7fe61f91bf155b4d6c11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.496ex; height:2.843ex;" alt="{\displaystyle \chi :G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )\rightarrow \mathbb {C} ^{\times }}" loading="lazy"> auffassen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{\times }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdb696059a558aeee27138277c41d30fd9bd7a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{\times }}" loading="lazy"> kann als die <a href="Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine, lineare Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL_{1}(\mathbb {C} )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle GL_{1}(\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7465ac890265f66d2e9e223e761da72f4b9a24d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.951ex; height:2.843ex;" alt="{\displaystyle GL_{1}(\mathbb {C} )}" loading="lazy"> des eindimensionalen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">-<a href="Vektorraum" title="Vektorraum">Vektorraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> interpretiert werden. So erhält man aus dem ursprünglichen Dirichlet-Charakter eine eindimensionale Darstellung

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi :G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )\longrightarrow GL_{1}(\mathbb {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a277876ac55092e3ba6fc8338fad8a9762e3ff26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.74ex; height:2.843ex;" alt="{\displaystyle \chi :G(\mathbb {Q} (\mu _{m})/\mathbb {Q} )\longrightarrow GL_{1}(\mathbb {C} )}" loading="lazy">,</dd></dl>

mit deren Hilfe sich die dem Dirichlet-Charakter zugeordnete L-Reihe auch in der Form schreiben lässt:<a href="#cite_note-61">[59]</a>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)=\prod _{p\not \mid m}(1-\chi (\phi _{p})p^{-s})^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3b94feb83beb04e3e3caf6ad1106144126e9333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.345ex; height:6.009ex;" alt="{\displaystyle L(\chi ,s)=\prod _{p\not \mid m}(1-\chi (\phi _{p})p^{-s})^{-1}}" loading="lazy">.</dd></dl>

Ausgehend von dieser initialen Beobachtung sind Artinsche L-Reihen folgendermaßen definiert: Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fff2e26b0293c94eefaf381c9d62e7779d487fbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.004ex; height:2.843ex;" alt="{\displaystyle E/K}" loading="lazy"> eine endliche, galoissche Erweiterung von Zahlkörpern mit Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=G(E/K)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=G(E/K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1de315b056608128772b565a8ed7e4b46a85d5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.212ex; height:2.843ex;" alt="{\displaystyle G:=G(E/K)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein endlich-dimensionaler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">-Vektorraum und

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho :G\longrightarrow GL(V),\;\sigma \longmapsto \rho (\sigma ),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43b824bf06aa9794fe11c65993c84708f7041c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.16ex; height:2.843ex;" alt="{\displaystyle \rho :G\longrightarrow GL(V),\;\sigma \longmapsto \rho (\sigma ),}" loading="lazy"></dd></dl>

eine Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\rho ,V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c652e5ff39a9db4bd49b2e864089ca1574ab12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (\rho ,V)}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, also ein Gruppenhomomorphismus der Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> in die allgemeine, lineare Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle GL(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ad853a4ebab014e00fbd05a7a75beca186ed9a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.006ex; height:2.843ex;" alt="{\displaystyle GL(V)}" loading="lazy"> der Vektorraumautomorphismen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> ein von Null verschiedenes Primideal von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> ein Primideal von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">, das über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> liegt, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> ist ein Faktor in der Primidealzerlegung des Ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}{\mathcal {O}}_{E}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}{\mathcal {O}}_{E}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ffe9363ea9249c6d1169038a845a6a07d6eeeb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.5ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}{\mathcal {O}}_{E}}" loading="lazy"> des Ganzheitsrings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{E}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{E}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b06da2e997707d5c11cc8c31d1285146421aa030.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{E}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">. Die Zerlegungsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist die Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\sigma \in G\mid \sigma ({\mathfrak {P}})={\mathfrak {P}}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\sigma \in G\mid \sigma ({\mathfrak {P}})={\mathfrak {P}}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9444611096172c147b1be9f38bc46e32478bf4a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.345ex; height:2.843ex;" alt="{\displaystyle \{\sigma \in G\mid \sigma ({\mathfrak {P}})={\mathfrak {P}}\}}" loading="lazy"> der Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Man hat einen kanonischen, <a href="Surjektiv" class="mw-redirect" title="Surjektiv">surjektiven</a> Gruppenhomomorphismus

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}\longrightarrow G(k_{\mathfrak {P}}/k_{\mathfrak {p}}),\;\sigma \longrightarrow {\overline {\sigma }},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b5e89a4b9664c88f7bd635c82819ce726b5510c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.581ex; height:3.009ex;" alt="{\displaystyle G_{\mathfrak {P}}\longrightarrow G(k_{\mathfrak {P}}/k_{\mathfrak {p}}),\;\sigma \longrightarrow {\overline {\sigma }},}" loading="lazy"></dd></dl>

von der Zerlegungsgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> auf die Galoisgruppe der Erweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathfrak {P}}/k_{\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathfrak {P}}/k_{\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc5c9816b3a27352819b5f07b97f80ff3f3178d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.232ex; height:3.009ex;" alt="{\displaystyle k_{\mathfrak {P}}/k_{\mathfrak {p}}}" loading="lazy"> endlicher Restkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathfrak {P}}:={\mathcal {O}}_{E}/{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathfrak {P}}:={\mathcal {O}}_{E}/{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3c0def50540ca292c78f3eed3c36be4126f12f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.974ex; height:3.009ex;" alt="{\displaystyle k_{\mathfrak {P}}:={\mathcal {O}}_{E}/{\mathfrak {P}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathfrak {p}}:={\mathcal {O}}_{K}/{\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathfrak {p}}:={\mathcal {O}}_{K}/{\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eff624858ebd5d36a7c49f9ddb18dc9fb9af2de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.879ex; height:3.009ex;" alt="{\displaystyle k_{\mathfrak {p}}:={\mathcal {O}}_{K}/{\mathfrak {p}}}" loading="lazy">. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sigma }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\sigma }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39f67f9912ea9babd16a0a1adc622f9aa0ec9a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\overline {\sigma }}}" loading="lazy"> der Körperautomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathfrak {P}}\rightarrow k_{\mathfrak {P}},\;x\mapsto \sigma x\operatorname {mod} {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathfrak {P}}\rightarrow k_{\mathfrak {P}},\;x\mapsto \sigma x\operatorname {mod} {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63925a6f3b7c93fed6db3ec3ba8300eea7de775d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.594ex; height:2.843ex;" alt="{\displaystyle k_{\mathfrak {P}}\rightarrow k_{\mathfrak {P}},\;x\mapsto \sigma x\operatorname {mod} {\mathfrak {P}}}" loading="lazy">. Der <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathfrak {P}}\subseteq G_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle I_{\mathfrak {P}}\subseteq G_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1157cf590a081a7f9a8803c1a0c5dc91a555cfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.134ex; height:2.843ex;" alt="{\displaystyle I_{\mathfrak {P}}\subseteq G_{\mathfrak {P}}}" loading="lazy"> dieses surjektiven Gruppenhomomorphismus heißt die Trägheitsgruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und liefert den kanonischen Gruppenisomorphismus

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}/I_{\mathfrak {P}}{\stackrel {\sim }{\longrightarrow }}G(k_{\mathfrak {P}}/k_{\mathfrak {p}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75771a71b1f79c92e727ccdb0c4ac57f1d262ec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.872ex; height:3.509ex;" alt="{\displaystyle G_{\mathfrak {P}}/I_{\mathfrak {P}}{\stackrel {\sim }{\longrightarrow }}G(k_{\mathfrak {P}}/k_{\mathfrak {p}})}" loading="lazy">.</dd></dl>

Als Galoisgruppe einer Erweiterung endlicher Körper ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(k_{\mathfrak {P}}/k_{\mathfrak {p}})}">

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<annotation encoding="application/x-tex">{\displaystyle G(k_{\mathfrak {P}}/k_{\mathfrak {p}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8769108ee048441e02f78bb9b10fd7e6c97a3aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.868ex; height:3.009ex;" alt="{\displaystyle G(k_{\mathfrak {P}}/k_{\mathfrak {p}})}" loading="lazy"> <a href="Zyklische_Gruppe" title="Zyklische Gruppe">zyklisch</a> und wird vom Frobeniusautomorphismus

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<annotation encoding="application/x-tex">{\displaystyle k_{\mathfrak {P}}/k_{\mathfrak {p}}\longrightarrow k_{\mathfrak {P}}/k_{\mathfrak {p}},\;\alpha \mapsto \alpha ^{q_{\mathfrak {p}}},\;q_{\mathfrak {p}}:={\mathcal {N}}({\mathfrak {p}})=\#k_{\mathfrak {p}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39671f2b099b4f5c3b4613726efd8596326d62bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.335ex; height:3.176ex;" alt="{\displaystyle k_{\mathfrak {P}}/k_{\mathfrak {p}}\longrightarrow k_{\mathfrak {P}}/k_{\mathfrak {p}},\;\alpha \mapsto \alpha ^{q_{\mathfrak {p}}},\;q_{\mathfrak {p}}:={\mathcal {N}}({\mathfrak {p}})=\#k_{\mathfrak {p}},}" loading="lazy"></dd></dl>

erzeugt. Jedes seiner <a href="Urbild_(Mathematik)" title="Urbild (Mathematik)">Urbilder</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> unter dem kanonischen Gruppenisomorphismus erzeugt die <a href="Faktorgruppe" title="Faktorgruppe">Faktorgruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}/I_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}/I_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d99b9e0dc8f066f1ecb7c562a13658254ff46027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.198ex; height:3.009ex;" alt="{\displaystyle G_{\mathfrak {P}}/I_{\mathfrak {P}}}" loading="lazy"> und wird ein Frobeniusautomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e88ee280fa274b42ee37c67f20e43c15148582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.978ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathfrak {P}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> genannt. Ist die Trägheitsgruppe nicht-trivial, so gibt es mehrere, solche Urbilder. Beschränkt man aber ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e88ee280fa274b42ee37c67f20e43c15148582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.978ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathfrak {P}}}" loading="lazy"> auf den Fixmodul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{I_{\mathfrak {P}}}:=\{v\in V\mid \forall \sigma \in I_{\mathfrak {P}}:\,\rho (\sigma )(v)=v\}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{I_{\mathfrak {P}}}:=\{v\in V\mid \forall \sigma \in I_{\mathfrak {P}}:\,\rho (\sigma )(v)=v\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b569dba5bab87540033afc175addaf65c2a4e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.811ex; height:3.343ex;" alt="{\displaystyle V^{I_{\mathfrak {P}}}:=\{v\in V\mid \forall \sigma \in I_{\mathfrak {P}}:\,\rho (\sigma )(v)=v\}}" loading="lazy">, einen <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, so entsteht unabhängig von der Wahl von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e88ee280fa274b42ee37c67f20e43c15148582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.978ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathfrak {P}}}" loading="lazy"> ein wohldefinierter <a href="Automorphismus" title="Automorphismus">Automorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathfrak {P}}:V^{I_{\mathfrak {P}}}\rightarrow V^{I_{\mathfrak {P}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{\mathfrak {P}}:V^{I_{\mathfrak {P}}}\rightarrow V^{I_{\mathfrak {P}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a07bcf7b6848913d63d1557a5f03de88a948d787.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.813ex; height:3.343ex;" alt="{\displaystyle \phi _{\mathfrak {P}}:V^{I_{\mathfrak {P}}}\rightarrow V^{I_{\mathfrak {P}}}}" loading="lazy">. Es bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\mathfrak {P}}:=\operatorname {dim} V^{I_{\mathfrak {P}}}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{\mathfrak {P}}:=\operatorname {dim} V^{I_{\mathfrak {P}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d7c023709d326aec8b139f2ef571a5343f0e5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.951ex; height:3.343ex;" alt="{\displaystyle d_{\mathfrak {P}}:=\operatorname {dim} V^{I_{\mathfrak {P}}}}" loading="lazy"> die <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> des Untervektorraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{I_{\mathfrak {P}}}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{I_{\mathfrak {P}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2e5e158559612af54e587d0a55313d792d1d6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.142ex; height:2.676ex;" alt="{\displaystyle V^{I_{\mathfrak {P}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/587ff3e66013dc060d685633c362bf6e0c9788e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.308ex; height:2.843ex;" alt="{\displaystyle E_{\mathfrak {P}}}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\mathfrak {P}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle d_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/214f72a10702416bea02f4ec6dcc29c71d742856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.802ex; height:2.843ex;" alt="{\displaystyle d_{\mathfrak {P}}}" loading="lazy">-dimensionale <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>.<a href="#cite_note-62">[60]</a>

Die Artinsche L-Reihe zur Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,V)}">

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<annotation encoding="application/x-tex">{\displaystyle (\rho ,V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c652e5ff39a9db4bd49b2e864089ca1574ab12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (\rho ,V)}" loading="lazy"> der Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=G(E/K)}">

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<annotation encoding="application/x-tex">{\displaystyle G=G(E/K)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9234fb20253a2f559f02383805fd149bce25db92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.565ex; height:2.843ex;" alt="{\displaystyle G=G(E/K)}" loading="lazy"> ist nun definiert durch<a href="#cite_note-63">[61]</a>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle L(\rho ,s):=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}\operatorname {det} (1-\phi _{\mathfrak {P}}q_{\mathfrak {p}}^{-s};V^{I_{\mathfrak {P}}})^{-1}:=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\big (}\operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)|_{X=q_{\mathfrak {p}}^{-s}}{\big )}^{-1},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ccf59ffc05bdf2a7aeb714544bb0f98fb0bac7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:79.383ex; height:6.176ex;" alt="{\displaystyle L(\rho ,s):=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}\operatorname {det} (1-\phi _{\mathfrak {P}}q_{\mathfrak {p}}^{-s};V^{I_{\mathfrak {P}}})^{-1}:=\prod _{0\neq {\mathfrak {p}}\subset {\mathcal {O}}_{K}}{\big (}\operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)|_{X=q_{\mathfrak {p}}^{-s}}{\big )}^{-1},}" loading="lazy"></dd></dl>

in der das Produkt alle von Null verschiedenen Primideale des Ganzheitsrings von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> durchläuft. Das erste Produkt ist die Kurzschreibweise dieser Definition, welche durch das zweite Produkt erläutert werden soll: Der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79daa3a201b366cffd988b7d6eafd90523adde13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.146ex; height:3.009ex;" alt="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}" loading="lazy"> liefert ein <a href="Polynom" title="Polynom">Polynom</a> vom <a href="Grad_(Polynom)" title="Grad (Polynom)">Grad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/214f72a10702416bea02f4ec6dcc29c71d742856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.802ex; height:2.843ex;" alt="{\displaystyle d_{\mathfrak {P}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [X]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [X]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6c0e47242946827f33aad1803d66106d2f0307e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [X]}" loading="lazy">. Im Vergleich zum <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristischen Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (XE_{\mathfrak {P}}-\phi _{\mathfrak {P}})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (XE_{\mathfrak {P}}-\phi _{\mathfrak {P}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fba6164348891d5a88919747da3cc6de9e2d3f1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.146ex; height:3.009ex;" alt="{\displaystyle \operatorname {det} (XE_{\mathfrak {P}}-\phi _{\mathfrak {P}})}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\mathfrak {P}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \phi _{\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e88ee280fa274b42ee37c67f20e43c15148582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.978ex; height:2.843ex;" alt="{\displaystyle \phi _{\mathfrak {P}}}" loading="lazy"> ist in diesem Polynom die Reihenfolge der <a href="Koeffizient" title="Koeffizient">Koeffizienten</a> umgekehrt. Die Schreibweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |_{X=q_{\mathfrak {p}}^{-s}}}">

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<annotation encoding="application/x-tex">{\displaystyle |_{X=q_{\mathfrak {p}}^{-s}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba019d04f08505b8d25d792a5d94f38e5181f11a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:6.149ex; height:3.676ex;" alt="{\displaystyle |_{X=q_{\mathfrak {p}}^{-s}}}" loading="lazy"> bedeutet, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\mathfrak {p}}^{-s}}">

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<annotation encoding="application/x-tex">{\displaystyle q_{\mathfrak {p}}^{-s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16bcba93a7fbd9bb0e9f2e8e6364f88922373581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.361ex; height:3.343ex;" alt="{\displaystyle q_{\mathfrak {p}}^{-s}}" loading="lazy"> an die Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in das Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79daa3a201b366cffd988b7d6eafd90523adde13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.146ex; height:3.009ex;" alt="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}" loading="lazy"> eingesetzt werden soll. Die einzelnen Euler-Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)|_{X=q_{\mathfrak {p}}^{-s}}{\big )}^{-1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (\operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)|_{X=q_{\mathfrak {p}}^{-s}}{\big )}^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35dbd399ea6c4635faa3d5dbea28240472e5e9a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:26.597ex; height:4.343ex;" alt="{\displaystyle (\operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)|_{X=q_{\mathfrak {p}}^{-s}}{\big )}^{-1}}" loading="lazy"> hängen nicht von der Wahl des über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> gelegenen Primideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> ab, da eine andere Wahl lediglich zu einem bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> konjugierten Frobeniusautomorphismus führt, welcher das Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79daa3a201b366cffd988b7d6eafd90523adde13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.146ex; height:3.009ex;" alt="{\displaystyle \operatorname {det} (E_{\mathfrak {P}}-\phi _{\mathfrak {P}}X)}" loading="lazy"> nicht ändert. Artinsche L-Reihen sind somit wohldefiniert. Sie konvergieren in der Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> absolut und für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

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<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> gleichmäßig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)\geq 1+\delta }">

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<annotation encoding="application/x-tex">{\displaystyle \Re (s)\geq 1+\delta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2594c21220ce85205a3503c907c32a30d80e8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.974ex; height:2.843ex;" alt="{\displaystyle \Re (s)\geq 1+\delta }" loading="lazy">. Sie stellen dort also <a href="Analytische_Funktion" title="Analytische Funktion">analytische (holomorphe) Funktionen</a> dar.<a href="#cite_note-64">[62]</a>

Der Charakter einer Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :G\mapsto GL(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \rho :G\mapsto GL(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d071b83d38da7a69f6ca18e53a8db96661c86510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.586ex; height:2.843ex;" alt="{\displaystyle \rho :G\mapsto GL(V)}" loading="lazy"> der Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> wird definiert als die Funktion

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<annotation encoding="application/x-tex">{\displaystyle \chi :=\chi _{\rho }:G\longrightarrow \mathbb {C} ,\sigma \longmapsto \operatorname {Spur} \rho (\sigma ).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e0fc390e46850c328f5153fda48060e2d7cf3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.899ex; height:3.009ex;" alt="{\displaystyle \chi :=\chi _{\rho }:G\longrightarrow \mathbb {C} ,\sigma \longmapsto \operatorname {Spur} \rho (\sigma ).}" loading="lazy"></dd></dl>

Zwei Darstellungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,V)}">

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<annotation encoding="application/x-tex">{\displaystyle (\rho ,V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c652e5ff39a9db4bd49b2e864089ca1574ab12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (\rho ,V)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ',V')}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\rho ',V')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b50cc696c64d031598bd6c8c4e6907e897808f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.332ex; height:3.009ex;" alt="{\displaystyle (\rho ',V')}" loading="lazy"> der Galoisgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> heißen äquivalent, wenn die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Moduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff31fe992a31b4c954a933f1be91e2739a1c0ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}" loading="lazy">, also die abelschen Gruppen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff31fe992a31b4c954a933f1be91e2739a1c0ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}" loading="lazy">, auf denen die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als Automorphismengruppe <a href="Gruppenoperation" title="Gruppenoperation">operiert</a>, isomorph sind. Dies ist genau dann der Fall, wenn ihre Charaktere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\rho }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{\rho }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a85b3961ece00c5b8c11b99e15681289526dadb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.537ex; height:2.343ex;" alt="{\displaystyle \chi _{\rho }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\rho '}}">

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{\rho '}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18216862954764c7264541ad92703fbfe14a7b23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.069ex; height:2.343ex;" alt="{\displaystyle \chi _{\rho '}}" loading="lazy"> gleich sind.<a href="#cite_note-65">[63]</a> Deshalb bezeichnet man die Artinsche L-Reihe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\rho ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\rho ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af5c2d6589fcaf15ef4fafefea514a84ebbb8838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle L(\rho ,s)}" loading="lazy"> in der Regel mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df2ede56cb317a2561bcfa48a1ea433b274781bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle L(\chi ,s)}" loading="lazy">.<a href="#cite_note-66">[64]</a> Entsprechend ihrer Definition durchläuft sie nur die endlichen Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, nämlich die von Null verschiedenen Primideale von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy">. Es fehlt noch der Gamma-Faktor an den unendlichen Stellen:

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (\chi ,s):=\prod _{{\mathfrak {p}}|\infty }L_{\mathfrak {p}}(\chi ,s),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a6c0d1abb93240588ef456ca22b8b1066012ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.426ex; height:6.009ex;" alt="{\displaystyle \gamma (\chi ,s):=\prod _{{\mathfrak {p}}|\infty }L_{\mathfrak {p}}(\chi ,s),}" loading="lazy"></dd></dl>

wobei

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</msup>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mrow>

<mtext>reell</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mtext>falls</mtext>

</mrow>

</mrow>

<mtext>komplex</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{\mathfrak {p}}(\chi ,s)={\begin{cases}\Gamma _{\mathbb {R} }(s)^{\operatorname {dim} V^{G_{\mathfrak {P}}}}\cdot \Gamma _{\mathbb {R} }(s+1)^{\operatorname {dim} V/V^{G_{\mathfrak {P}}}}&{\text{falls}}\;{\mathfrak {p}}\;{\text{reell}}\\\Gamma _{\mathbb {C} }(s)^{\operatorname {dim} V}&{\text{falls}}\;{\mathfrak {p}}\;{\text{komplex}}\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f07b06a1e96c6fc6ce13ae64f4f60991c456f9ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.776ex; height:7.509ex;" alt="{\displaystyle L_{\mathfrak {p}}(\chi ,s)={\begin{cases}\Gamma _{\mathbb {R} }(s)^{\operatorname {dim} V^{G_{\mathfrak {P}}}}\cdot \Gamma _{\mathbb {R} }(s+1)^{\operatorname {dim} V/V^{G_{\mathfrak {P}}}}&{\text{falls}}\;{\mathfrak {p}}\;{\text{reell}}\\\Gamma _{\mathbb {C} }(s)^{\operatorname {dim} V}&{\text{falls}}\;{\mathfrak {p}}\;{\text{komplex}}\end{cases}}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\mathbb {R} }(z):=\pi ^{-z/2}\Gamma (z/2)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mathbb {R} }(z):=\pi ^{-z/2}\Gamma (z/2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6b0cc12b824885b5303277f673391b3e2d4413e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.448ex; height:3.343ex;" alt="{\displaystyle \Gamma _{\mathbb {R} }(z):=\pi ^{-z/2}\Gamma (z/2)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\mathbb {C} }(z):=2\cdot (2\pi )^{-z}\Gamma (z)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mathbb {C} }(z):=2\cdot (2\pi )^{-z}\Gamma (z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdbfafb3bdfdbebec4e1b3b647eed1dac6f3547e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.29ex; height:3.009ex;" alt="{\displaystyle \Gamma _{\mathbb {C} }(z):=2\cdot (2\pi )^{-z}\Gamma (z)}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> als der Zerlegungsgruppe einer über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> gelegenen Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy">. Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> reell, so besteht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> nur aus der Identität und ist somit einelementig. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> reell, aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> komplex, so besitzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathfrak {P}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455879ea5974ae11cdfe70eb2256d0facbb9e3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.42ex; height:2.843ex;" alt="{\displaystyle G_{\mathfrak {P}}}" loading="lazy"> zwei Elemente: die Identität und das eindeutige Element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, welches die beiden komplexen Einbettungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">, die zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f306de895b1fd787a364c508badc106b1ff73e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.924ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {P}}}" loading="lazy"> gehören, vertauscht.<a href="#cite_note-67">[65]</a><a href="#cite_note-68">[66]</a>

Der Artin-Führer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}(\chi )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}(\chi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0c408cf706a3b24557a8a772a93e9555b02c8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.023ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {f}}(\chi )}" loading="lazy"> eines Charakters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> einer Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist ein Ideal von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, welches die Verzweigung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> misst.<a href="#cite_note-69">[67]</a> Mit seiner Hilfe lässt sich nun die vervollständigte Artinsche L-Funktion definieren durch

<msup>

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</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

<msub>

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</msub>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

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</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s):=q(\chi )^{s/2}\cdot \gamma (\chi ,s)\cdot L(\chi ,s)=(|d_{K}|^{\operatorname {dim} V}{\mathcal {N}}{\mathfrak {f}}(\chi ))^{s/2}\prod _{{\mathfrak {p}}\mid \infty }L_{\mathfrak {p}}(\chi ,s)\cdot \prod _{{\mathfrak {p}}\not \mid \infty }(\operatorname {det} (1-\phi _{\mathfrak {P}}q_{\mathfrak {p}}^{-s};V^{I_{\mathfrak {P}}})^{-1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02ecb599df5b9dcc4bbe2a8d693e4fd6c78495a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:98.24ex; height:6.009ex;" alt="{\displaystyle \Lambda (\chi ,s):=q(\chi )^{s/2}\cdot \gamma (\chi ,s)\cdot L(\chi ,s)=(|d_{K}|^{\operatorname {dim} V}{\mathcal {N}}{\mathfrak {f}}(\chi ))^{s/2}\prod _{{\mathfrak {p}}\mid \infty }L_{\mathfrak {p}}(\chi ,s)\cdot \prod _{{\mathfrak {p}}\not \mid \infty }(\operatorname {det} (1-\phi _{\mathfrak {P}}q_{\mathfrak {p}}^{-s};V^{I_{\mathfrak {P}}})^{-1}.}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(\chi ):=|d_{K}|^{\operatorname {dim} V}{\mathcal {N}}{\mathfrak {f}}(\chi )}">

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(\chi ):=|d_{K}|^{\operatorname {dim} V}{\mathcal {N}}{\mathfrak {f}}(\chi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a68225d0b66aa471063d037baa4d346fbccc3b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.196ex; height:3.343ex;" alt="{\displaystyle q(\chi ):=|d_{K}|^{\operatorname {dim} V}{\mathcal {N}}{\mathfrak {f}}(\chi )}" loading="lazy"> der Führer der L-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> , welcher eine Potenz des Betrags der Diskriminante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{K}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20e051a048aaa0ee9c8c5d241bb54a9d768a811b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.902ex; height:2.509ex;" alt="{\displaystyle d_{K}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und die Norm des Artin-Führers von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> enthält. Diese Definition der vervollständigten Artinschen L-Funktion ist zunächst nur in der Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> gültig. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> kann aber meromorph auf die ganze komplexe Zahlenebene fortgesetzt werden. Bezeichnet man auch diese fortgesetzte L-Funktion mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy">, so erfüllt sie die Funktionalgleichung

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,1-s)=\epsilon (\chi )\Lambda ({\overline {\chi }},s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad800d6f1f2f408e98a17a6d42c33ede20a736e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.428ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,1-s)=\epsilon (\chi )\Lambda ({\overline {\chi }},s)}" loading="lazy"></dd></dl>

mit einer komplexen Konstanten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (\chi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon (\chi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d7714f34385d79d40bb67f4ce3fbcc672f66ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle \epsilon (\chi )}" loading="lazy">, welche den Absolutbetrag 1 besitzt und die Artinsche Wurzelzahl von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\chi ,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda (\chi ,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce994bcc8248c1303bebc1da706f526c32d68284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.002ex; height:2.843ex;" alt="{\displaystyle \Lambda (\chi ,s)}" loading="lazy"> genannt wird.<a href="#cite_note-70">[68]</a> Der Grad von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/695dbc99932d97d01154b849d305dbdaebe6cdb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.05ex; height:2.176ex;" alt="{\displaystyle \operatorname {dim} V}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} V\cdot [K:\mathbb {Q} ]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} V\cdot [K:\mathbb {Q} ]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5815cff50fb00a12f54c0d4090d69e0ba81933a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.834ex; height:2.843ex;" alt="{\displaystyle \operatorname {dim} V\cdot [K:\mathbb {Q} ]}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">.<a href="#cite_note-71">[69]</a>

Die bislang unbewiesene Artin-Vermutung besagt, jede Artinsche L-Funktion lasse sich analytisch (holomorph) von der Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> fortsetzen, sofern der Charakter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> bei der Zerlegung in Charaktere zu irreduziblen Darstellungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> keinen trivialen Charakter enthält. Diese Vermutung gilt z. B. für eindimensionale Darstellungen, ist aber im allgemeinen Fall unbewiesen.<a href="#cite_note-72">[70]</a> Die Existenz von Artinschen L-Funktionen, die im kritischen Streifen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\Re (s)<1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\Re (s)<1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f3c5ef2ecd43a7d778852632aba5355e5838d2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.346ex; height:2.843ex;" alt="{\displaystyle 0<\Re (s)<1}" loading="lazy"> Polstellen besitzen, kann also bisher nicht ausgeschlossen werden.<a href="#cite_note-73">[71]</a>

Artinsche L-Funktionen lassen sich nicht nur im Bezug auf Darstellungen von endlichen Galoiserweiterungen algebraischer Zahlkörper definieren. Allgemeiner kann man <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globale Körper</a> zu Grunde legen, zu denen die algebraischen Zahlkörper gehören. Neben den Fall „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6e1cfbaa4da6cb13b2f00615638a7065ce0f0a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.164ex; height:2.176ex;" alt="{\displaystyle K=\;}" loading="lazy">algebraischer Zahlkörper“ tritt dann auch der Fall „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {F} _{p}(T)=\;}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {F} _{p}(T)=\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01fb102a390b046c1e0e231ae73aad696d969634.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.188ex; height:3.009ex;" alt="{\displaystyle K=\mathbb {F} _{p}(T)=\;}" loading="lazy"><a href="Funktionenk%C3%B6rper" title="Funktionenkörper">Funktionenkörper</a> einer Variablen über einem endlichen Körper mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy"> vielen Elementen.“<a href="#cite_note-74">[72]</a>

<div class="mw-heading mw-heading2"><h2 id="Vermutete_Eigenschaften">Vermutete Eigenschaften</h2></div>

Man kann aus bekannten Beispielen ablesen, welche Eigenschaften eine Theorie der L-Funktionen haben sollte, und zwar sollte sie

<ol><li>die Position der Null- und Polstellen ergeben,</li>

<li>Funktionalgleichungen bezüglich der Vertikallinien Re (s) = constant liefern,</li>

<li>spezielle und interessante Werte für ganzzahlige Argumente ergeben.</li></ol>

Detailuntersuchungen haben eine große Zahl plausibler Vermutungen erzeugt, zum Beispiel über den genauen Typ der gerade angegebenen Funktionalgleichungen. Da die <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannsche ζ-Funktion</a> durch ihre Werte bei geradzahligen positiven ganzen Zahlen (und negativen ungeradzahligen Werten) mit den <a href="Bernoulli-Zahl" title="Bernoulli-Zahl">Bernoullischen Zahlen</a> zusammenhängt, liegt es nahe, nach einer Verallgemeinerung der Bernoullischen Zahlen in der angegebenen Theorie zu suchen. Man verwendet dazu die Körper der <a href="P-adische_Zahlen" class="mw-redirect" title="P-adische Zahlen">p-adischen Zahlen</a>, wodurch gewisse Galois-Moduln beschrieben werden.

Die <a href="Statistik" title="Statistik">statistischen Eigenschaften</a> der Nullstellenverteilung der L-Funktionen sind unter anderen deshalb von Interesse, weil sie mit allgemeinen Problemen zusammenhängt, zum Beispiel mit einer Hypothese über die Primzahlverteilung und mit anderen sogenannten verallgemeinerten Riemannschen Hypothesen. Der Zusammenhang mit den Theorien der <a href="Zufallsmatrix" title="Zufallsmatrix">Zufallsmatrizen</a> und des sogenannten <a href="Quantenchaos" title="Quantenchaos">Quantenchaos</a> ist ebenfalls von Interesse. Die <a href="Fraktal" title="Fraktal">fraktale Struktur</a> der Verteilungen wurde ebenfalls mit sogenannten Skalenanalysen untersucht.<a href="#cite_note-Shanker-75">[73]</a> Die Selbstähnlichkeit der Nullstellenverteilung ist sehr bemerkenswert und wird durch einen großen Wert der <a href="Fraktale_Dimension" title="Fraktale Dimension">fraktalen Dimension</a>, ~ 1.9, charakterisiert. Dieser sehr hohe Wert gilt für mehr als 15 Größenordnungen der Nullstellenverteilung der Riemannschen ζ-Funktion und auch für die Nullstellen anderer L-Funktionen.

<a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> fasste 1991 die vermuteten oder bewiesenen Eigenschaften vieler Zeta- und L-Funktionen in seiner axiomatischen Definition der <a href="Selberg-Klasse" title="Selberg-Klasse">Selberg-Klasse</a> zusammen.<a href="#cite_note-76">[74]</a><a href="#cite_note-77">[75]</a> Die vier Axiome sind Analytizität, Eulerprodukt, Ramanujan-Vermutung, Funktionalgleichung. Selberg stellte dann weitere Vermutungen über die Selberg-Klasse auf.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Tom_M._Apostol" class="mw-redirect" title="Tom M. Apostol">Tom M. Apostol</a>: <cite style="font-style:italic">Introduction to Analytic Number Theory</cite>. Springer, New York 1976, ISBN 0-387-90163-9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.au=Tom+M.+Apostol&rft.btitle=Introduction+to+Analytic+Number+Theory&rft.date=1976&rft.genre=book&rft.isbn=0387901639&rft.place=New+York&rft.pub=Springer" style="display:none"> </li>

<li><a href="Peter_Borwein" title="Peter Borwein">Peter Borwein</a>, Stephen Choi, Brendan Rooney, Andrea Weirathmueller: <cite style="font-style:italic">The Riemann Hypothesis</cite>. Springer, New York 2008, ISBN 978-0-387-72125-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.au=Peter+Borwein%2C+Stephen+Choi%2C+Brendan+Rooney%2C+...&rft.btitle=The+Riemann+Hypothesis&rft.date=2008&rft.genre=book&rft.isbn=9780387721255&rft.place=New+York&rft.pub=Springer" style="display:none"> </li>

<li><a href="Daniel_Bump" title="Daniel Bump">Daniel Bump</a> u. a.: An Introduction to the Langlands Program. Herausgeber: <a href="Joseph_Bernstein" title="Joseph Bernstein">Joseph Bernstein</a>, <a href="Stephen_Gelbart" title="Stephen Gelbart">Stephen Gelbart</a>. Birkhäuser, Boston 2004, ISBN 3-7643-3211-5.</li>

<li><a href="Henri_Cohen_(Mathematiker)" title="Henri Cohen (Mathematiker)">Henri Cohen</a>: Advanced Topics in Computational Number Theory (= Graduate Texts in Mathematics. Band 193). 1. Auflage. Springer-Verlag, New York 2000, ISBN 0-387-98727-4. (Insbesondere Kapitel 3 und 4 sowie Abschnitt 10.3)</li>

<li>Ze Li Dou, Qiao Zhang: Six Short Chapters on Automorphic Forms and L-Functions. Springer, Berlin, Heidelberg 2012, ISBN 978-3-642-28708-4.</li>

<li><a href="Stephen_Gelbart" title="Stephen Gelbart">Stephen Gelbart</a>, lIya Piatetski-Shapiro, <a href="Stephen_Rallis" title="Stephen Rallis">Stephen Rallis</a>: Explicit Constructions of Automorphic L-Functions (= Lecture Notes in Mathematics. Band 1254). Springer-Verlag, Berlin, Heidelberg, New York 1987, ISBN 3-540-17848-1.</li>

<li><a href="Haruzo_Hida" title="Haruzo Hida">Haruzo Hida</a>: Elementary Theory of L-functions and Eisenstein Series (= London Mathematical Society Student Texts. Band 26). Cambridge University Press, Cambridge 1993, ISBN 978-0-521-43411-9.</li>

<li><a href="Aleksandar_Ivi%C4%87" title="Aleksandar Ivić">Aleksandar Ivić</a>: <cite style="font-style:italic">The Riemann Zeta-Function: theory and applications</cite>. Dover, Mineola 2003, ISBN 0-486-42813-3.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.au=Aleksandar+Ivi%C4%87&rft.btitle=The+Riemann+Zeta-Function%3A+theory+and+applications&rft.date=2003&rft.genre=book&rft.isbn=0486428133&rft.place=Mineola&rft.pub=Dover" style="display:none"> </li>

<li><a href="Henryk_Iwaniec" title="Henryk Iwaniec">Henryk Iwaniec</a>, <a href="Emmanuel_Kowalski" title="Emmanuel Kowalski">Emmanuel Kowalski</a>: <cite style="font-style:italic">Analytic Number Theory</cite>. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, Providence 2004, ISBN 0-8218-3633-1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.au=Henryk+Iwaniec%2C+Emmanuel+Kowalski&rft.btitle=Analytic+Number+Theory&rft.date=2004&rft.genre=book&rft.isbn=0821836331&rft.place=Providence&rft.pub=American+Mathematical+Society" style="display:none"> </li>

<li><a href="Kenkichi_Iwasawa" class="mw-redirect" title="Kenkichi Iwasawa">Kenkichi Iwasawa</a>: Hecke's L-Functions - Spring, 1964. SpringerBriefs in Mathematics, Springer Singapore 2019, ISBN 978-981-13-9495-9.</li>

<li><a href="Bruno_Kahn" title="Bruno Kahn">Bruno Kahn</a>: Zeta and L-Functions of Varieties and Motives (= London Mathematical Society Lecture Note Series. Band 462). Cambridge University Press, 2020, ISBN 978-1-108-70339-0.</li>

<li><a href="Anatoli_Karazuba" class="mw-redirect" title="Anatoli Karazuba">Anatoly A. Karatsuba</a>, <a href="Sergei_Michailowitsch_Woronin" title="Sergei Michailowitsch Woronin">S. M. Voronin</a>: <cite style="font-style:italic">The Riemann Zeta-Function</cite>. Walter de Gruyter, Berlin 1992, ISBN 3-11-013170-6.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.au=Anatoly+A.+Karatsuba%2C+S.+M.+Voronin&rft.btitle=The+Riemann+Zeta-Function&rft.date=1992&rft.genre=book&rft.isbn=3110131706&rft.place=Berlin&rft.pub=Walter+de+Gruyter" style="display:none"> </li>

<li><a href="Winnie_Li" title="Winnie Li">Wen-Ching Winnie Li</a>: Zeta and L-functions in number theory and combinatorics (= CBMS regional conference series in mathematics. Band 129). American Mathematical Society, 2019, ISBN 978-1-4704-4900-1.</li>

<li><a href="J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Jürgen Neukirch</a>: Algebraische Zahlentheorie. Springer-Verlag, Berlin/Heidelberg 1992, ISBN 3-540-54273-6 (insbesondere Kapitel VII).</li>

<li>Alberto Perelli: An Introduction to the Selberg Class of L-Functions. Vortragsskript, Vilnius Universität, Ph. D. Summer School in Number Theory and Probability, Druskininkai, Litauen, September 2007, <a rel="nofollow" class="external text" href="http://www.mif.vu.lt/katedros/ttsk/vs2007/files/Selberg.pdf">Link</a>.</li>

<li>Cristian Popescu, <a href="Karl_Rubin" title="Karl Rubin">Karl Rubin</a>, <a href="Alice_Silverberg" title="Alice Silverberg">Alice Silverberg</a> (Hrsg.): Arithmetic of L-functions (= IAS/Park City Mathematics Series. Band 18). 1. Auflage. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, Providence, Rhode Island, USA 2011, ISBN 978-0-8218-5320-7.</li>

<li><a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a>: Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Università di Salerno, 1992, S. 367–385. Auch enthalten in: Collected Papers II / Atle Selberg, Springer Collected Works in Mathematics (SCWM), Springer Berlin, Heidelberg 1991, S. 47–64, ISBN 978-3-642-41022-2.</li>

<li><a href="J%C3%B6rn_Steuding" title="Jörn Steuding">Jörn Steuding</a>: Value-Distribution of L-Functions (= Lecture Notes in Mathematics. Band 1877). 1. Auflage. Springer, Berlin, Heidelberg, New York 2007, ISBN 978-3-540-26526-9.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.lmfdb.org/">The L-functions and modular forms database - LMFDB</a> Umfangreiche Sammlung von L-Funktionen und anderer, zahlentheoretischer Objekte.</li>

<li><a rel="nofollow" class="external text" href="http://www.physorg.com/news124636003.html">Glimpses of a new (mathematical) world</a>, Physorg.com, 13. März 2008.</li>

<li><a rel="nofollow" class="external text" href="http://www.physorg.com/news137248087.html">Hunting the elusive L-function</a>.</li>

<li><a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/0000-000-00/S0273-0979-2018-01646-7/S0273-0979-2018-01646-7.pdf">Analytic L-functions: Definitions, Theorems and Connections</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Atle Selberg: Old and new conjectures and results about a class of Dirichlet series. 1989.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 1, S. 94.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 1, S. 95.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a>: Old and new conjectures and results about a class of Dirichlet-series. In: <a href="Enrico_Bombieri" title="Enrico Bombieri">Enrico Bombieri</a> u. a. (Hrsg.): Proceedings of the Amalfi Conference on Analytic Number Theory. 1992, S. 367–385; Collected Papers. Vol. II, Springer, 1991, S. 47–63.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 1, 1992, S. 439 ff.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 1, Satz 1.1, 1992, S. 439.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 1, Theorem 1.6, 1992, S. 445.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 1, Theorem 1.6, 1992, S. 445.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 1, Korollar 1.7, 1992, S. 446.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, 1992, S. 454 f.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> <a href="Tom_M._Apostol" class="mw-redirect" title="Tom M. Apostol">Tom M. Apostol</a>: Note on the trivial zeros of Dirichlet L-functions. In: Proceedings of the American Mathematical Society. Band 94, Nummer 1, S. 29–30. <a href="https://doi.org/10.1090/S0002-9939-1985-0781049-8" class="extiw external" title="doi:10.1090/S0002-9939-1985-0781049-8">doi:10.1090/S0002-9939-1985-0781049-8</a>.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Satz 2.1, 1992, S. 455.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Satz 2.1, 1992, S. 455.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, 1992, S. 457.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Theorem 2.8, 1992, S. 461.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 9, S. 119.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Definition 2.5, 1992, S. 459.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Theorem 2.8, 1992, S. 461.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, Satz 2.6, 1992, S. 459, Theorem 2.8, S. 461.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 2, 1992, S. 455.

</li>

<li id="cite_note-22"><a href="#cite_ref-22">↑</a> <a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">P. G. L. Dirichlet</a>: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. In: Abhand. Ak. Wiss. Berlin. (1837), S. 45–81; Werke I (1889), S. 313–342.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Satz 5.14, 1992, S. 490.

</li>

<li id="cite_note-24"><a href="#cite_ref-24">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Satz 5.13, 1992, S. 490.

</li>

<li id="cite_note-25"><a href="#cite_ref-25">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Definition 5.1, 1992, S. 478.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">↑</a> Steuding: Value-Distribution of L-Functions. 2007, Kapitel 13, Abschnitt 1, S. 250.

</li>

<li id="cite_note-27"><a href="#cite_ref-27">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Satz 5.2, 1992, S. 478.

</li>

<li id="cite_note-28"><a href="#cite_ref-28">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 10, S. 125.

</li>

<li id="cite_note-29"><a href="#cite_ref-29">↑</a> Steuding: Value-Distribution of L-Functions. 2007, Kapitel 13, Abschnitt 1, S. 250.

</li>

<li id="cite_note-30"><a href="#cite_ref-30">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 10, S. 125.

</li>

<li id="cite_note-31"><a href="#cite_ref-31">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 10, S. 125.

</li>

<li id="cite_note-32"><a href="#cite_ref-32">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Korollar 5.10, 1992, S. 487.

</li>

<li id="cite_note-33"><a href="#cite_ref-33">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 10, S. 125.

</li>

<li id="cite_note-34"><a href="#cite_ref-34">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Korollar 5.10, 1992, S. 487.

</li>

<li id="cite_note-35"><a href="#cite_ref-35">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, 1992, S. 488.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 5, Korollar 5.11, 1992, S. 488.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 6, S. 491.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 8, S. 515.

</li>

<li id="cite_note-39"><a href="#cite_ref-39">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 6, Definition 6.1, S. 492.

</li>

<li id="cite_note-40"><a href="#cite_ref-40">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 3, S. 464.

</li>

<li id="cite_note-41"><a href="#cite_ref-41">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 1, Paragraph 5, S. 31.

</li>

<li id="cite_note-42"><a href="#cite_ref-42">↑</a> Cohen: Advanced Topics in Computational Number Theory. 2000, Kapitel 3, Abschnitt 3, S. 135.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 6, S. 492.

</li>

<li id="cite_note-44"><a href="#cite_ref-44">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 6, S. 494.

</li>

<li id="cite_note-45"><a href="#cite_ref-45">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraphen 6 bis 8, S. 491–525.

</li>

<li id="cite_note-46"><a href="#cite_ref-46">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 377.

</li>

<li id="cite_note-47"><a href="#cite_ref-47">↑</a> Neukirch: Algebraische Zahlentheorie. 1992, Kapitel 7, Paragraph 8, Bemerkung 2, S. 525.

</li>

<li id="cite_note-49"><a href="#cite_ref-49">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 377.

</li>

<li id="cite_note-50"><a href="#cite_ref-50">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 378.

</li>

<li id="cite_note-51"><a href="#cite_ref-51">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 378 und 379.

</li>

<li id="cite_note-52"><a href="#cite_ref-52">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 382.

</li>

<li id="cite_note-53"><a href="#cite_ref-53">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 379.

</li>

<li id="cite_note-54"><a href="#cite_ref-54">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 383.

</li>

<li id="cite_note-55"><a href="#cite_ref-55">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 358, 383 und 384.

</li>

<li id="cite_note-56"><a href="#cite_ref-56">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 384, Abschnitt 3.2.

</li>

<li id="cite_note-57"><a href="#cite_ref-57">↑</a> Perelli: An Introduction to the Selberg Class of L-Functions. 2007, Kap. 3, Invariants, S. 17.

</li>

<li id="cite_note-58"><a href="#cite_ref-58">↑</a> Perelli: An Introduction to the Selberg Class of L-Functions. 2007, Kap. 2, Basic Theory of the Selberg Class, S. 7.

</li>

<li id="cite_note-59"><a href="#cite_ref-59">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 384, Theorem 2.1.

</li>

<li id="cite_note-60"><a href="#cite_ref-60">↑</a> Popescu et al. (Hrsg.): Arithmetic of L-Functions. 2011, Part III, Lecture 2, S. 384, Abschnitt 3.2, mit S. 361, Proposition 1.1.

</li>

<li id="cite_note-61"><a href="#cite_ref-61">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, Einleitung, 1992, S. 539.

</li>

<li id="cite_note-62"><a href="#cite_ref-62">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, 1992, S. 539 und 540.

</li>

<li id="cite_note-63"><a href="#cite_ref-63">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, Definition 10.1, 1992, S. 540.

</li>

<li id="cite_note-64"><a href="#cite_ref-64">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, 1992, S. 540.

</li>

<li id="cite_note-65"><a href="#cite_ref-65">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, 1992, S. 541.

</li>

<li id="cite_note-66"><a href="#cite_ref-66">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 10, 1992, S. 544.

</li>

<li id="cite_note-67"><a href="#cite_ref-67">↑</a> Ehud de Shalit: Artin L Functions. In: An Introduction to the Langlands Program. Bernstein, Gelbart (Hrsg.), 2004, Kapitel 4, Abschnitt 3.1, S. 75, und Abschnitt 4.1, S. 80.

</li>

<li id="cite_note-68"><a href="#cite_ref-68">↑</a> Neukirch: Algebraische Zahlentheorie. Kapitel 7, Paragraph 12, 1992, S. 558 f.

</li>

<li id="cite_note-69"><a href="#cite_ref-69">↑</a> Ehud de Shalit: Artin L Functions. In: An Introduction to the Langlands Program. Bernstein, Gelbart (Hrsg.), 2004, Kapitel 4, Abschnitt 4.2, S. 80.

</li>

<li id="cite_note-70"><a href="#cite_ref-70">↑</a> Ehud de Shalit: Artin L Functions. In: An Introduction to the Langlands Program. Bernstein, Gelbart (Hrsg.), 2004, Kapitel 4, Abschnitt 4.3, Theorem 4.1, S. 81.

</li>

<li id="cite_note-71"><a href="#cite_ref-71">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 4, Abschnitt 5.13, S. 141.

</li>

<li id="cite_note-72"><a href="#cite_ref-72">↑</a> Ehud de Shalit: Artin L Functions. In: An Introduction to the Langlands Program. Bernstein, Gelbart (Hrsg.), 2004, Kapitel 4, Conjecture 5.1, S. 83.

</li>

<li id="cite_note-73"><a href="#cite_ref-73">↑</a> Iwaniec, Kowalski: Analytic Number Theory. 2004, Kapitel 5, Abschnitt 5.13, S. 142.

</li>

<li id="cite_note-74"><a href="#cite_ref-74">↑</a> Ehud de Shalit: Artin L Functions. In: An Introduction to the Langlands Program. Bernstein, Gelbart (Hrsg.), 2004, Kapitel 4, Abschnitt 3.1, S. 75.

</li>

<li id="cite_note-Shanker-75"><a href="#cite_ref-Shanker_75-0">↑</a> O. Shanker: <cite style="font-style:italic">Random matrices, generalized zeta functions and self-similarity of zero distributions</cite>. In: <cite style="font-style:italic">J. Phys. A: Math. Gen.</cite> Band 39, 2006, S. 13983–13997, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088/0305-4470%2F39%2F45%2F008">10.1088/0305-4470/39/45/008</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:L-Funktion&rft.atitle=Random+matrices%2C+generalized+zeta+functions+and+self-similarity+of+zero+distributions&rft.au=O.+Shanker&rft.btitle=J.+Phys.+A%3A+Math.+Gen.&rft.date=2006&rft.doi=10.1088%2F0305-4470%2F39%2F45%2F008&rft.genre=book&rft.pages=13983-13997&rft.volume=39" style="display:none">

</li>

<li id="cite_note-76"><a href="#cite_ref-76">↑</a> Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Univ. Salerno, 1991, S. 367–385

</li>

<li id="cite_note-77"><a href="#cite_ref-77">↑</a> M. Ram Murty, Selberg's conjectures and Artin L-functions, Bull. AMS, Band 31, 1994, S. 1-14, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9407219">Arxiv</a>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2></div>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Die Ramanujan-Vermutung bezieht sich auf die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (f,n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (f,n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/081effc7550a2517d4a97da1f9605fbbfeff49c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.872ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (f,n)}" loading="lazy"> der Dirichlet-Reihe. Sie besagt: Für beliebiges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \epsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \epsilon >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dee909ae0cfd9a5d38d3483705f515e8df6807cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \textstyle \epsilon >0}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \lambda (f,n)=O(n^{\epsilon })}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \lambda (f,n)=O(n^{\epsilon })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a388fd90fdf108a2b9d218a413a7bfa54b833dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.847ex; height:2.843ex;" alt="{\displaystyle \textstyle \lambda (f,n)=O(n^{\epsilon })}" loading="lazy">. Dabei darf die implizite Konstante im <a href="Landau-Symbol" class="mw-redirect" title="Landau-Symbol">Landau-Symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle O}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle O}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f0bcba6024022faf53049354784ff309a6e3e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle \textstyle O}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \epsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \epsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7a607cfd7220d79ee384166b8b99ad931b00439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \textstyle \epsilon }" loading="lazy"> abhängen.

</li>

<li id="cite_note-48"><a href="#cite_ref-48">↑</a> Erläuterung der Begriffe „Stelle“ und „komplette Hülle“: Die Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> sind, bis auf Äquivalenz der Absolutbeträge, der gewöhnliche Absolutbetrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\infty }:=|\cdot |:\mathbb {Q} \rightarrow \mathbb {R} _{\geq 0}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\infty }:=|\cdot |:\mathbb {Q} \rightarrow \mathbb {R} _{\geq 0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a57edeeb6bd276f5135a8826c2cf58987736156e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.937ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{\infty }:=|\cdot |:\mathbb {Q} \rightarrow \mathbb {R} _{\geq 0}}" loading="lazy"> sowie die p-adischen Absolutbeträge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{p}\colon \mathbb {Q} \rightarrow \mathbb {R} _{\geq 0},|x|_{p}:=p^{-e_{p}(x)}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{p}\colon \mathbb {Q} \rightarrow \mathbb {R} _{\geq 0},|x|_{p}:=p^{-e_{p}(x)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39e7606f185a16866f34e8dbe02ef1fa2f36828f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.462ex; height:3.676ex;" alt="{\displaystyle |\cdot |_{p}\colon \mathbb {Q} \rightarrow \mathbb {R} _{\geq 0},|x|_{p}:=p^{-e_{p}(x)}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{p}(x)\in \mathbb {Z} }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{p}(x)\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3a190e45918bcf1ae6c06c9251d35e7b3665cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.673ex; height:3.009ex;" alt="{\displaystyle e_{p}(x)\in \mathbb {Z} }" loading="lazy"> den Exponenten der Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in der Primfaktorzerlegung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"> bezeichnet und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0|_{p}:=0}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |0|_{p}:=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19876e946aed8e8e29d527dc0151eeb8bb441de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.423ex; height:3.176ex;" alt="{\displaystyle |0|_{p}:=0}" loading="lazy"> gesetzt wird. Die entsprechenden kompletten Hüllen sind der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7126e801916f9176a090a967def39223e40d492b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.107ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} }" loading="lazy"> der reellen Zahlen sowie die <a href="P-adischer_K%C3%B6rper" class="mw-redirect" title="P-adischer Körper">p-adischen Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy">.

</li>

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