GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Selberg-Delange-Methode</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Selberg-Delange-Methode</span></h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Selberg-Delange-Methode</b> ist eine Technik aus der <a href="Analytische_Zahlentheorie" title="Analytische Zahlentheorie">analytischen Zahlentheorie</a>. Sie dient dazu, die mittlere Ordnung einer <a href="Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion">zahlentheoretischen Funktion</a> zu bestimmen. Sie ist nach <a href="Atle_Selberg" title="Atle Selberg">Atle Selberg</a> und <a href="Hubert_Delange" title="Hubert Delange">Hubert Delange</a> benannt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Die_Klassen_T_und_P">Die Klassen <i>T</i> und <i>P</i></h2></div>

<p>Seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} ,c_{0}>0,0<\delta \leq 1,M>0.}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} ,c_{0}>0,0<\delta \leq 1,M>0.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1a2a5375cbc3b9ea3dd394cb8f28aee3c6b7159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.951ex; height:2.676ex;" alt="{\displaystyle z\in \mathbb {C} ,c_{0}>0,0<\delta \leq 1,M>0.}" loading="lazy"></span> Ist nun <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle F(s)=\sum a(n)/n^{s}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle F(s)=\sum a(n)/n^{s}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8256e305de4da89e19c36208f261100021b67b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.574ex; height:2.843ex;" alt="{\displaystyle \textstyle F(s)=\sum a(n)/n^{s}}" loading="lazy"></span> eine <a href="Dirichlet-Reihe" class="mw-redirect" title="Dirichlet-Reihe">Dirichlet-Reihe</a> mit Konvergenzhalbebene <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82e22215d90eb4893d05f9b2bff409094a3451f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>1}" loading="lazy"></span>, so gehört diese zur Klasse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(z;c_{0},\delta ,M)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(z;c_{0},\delta ,M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f026d5513f35329d142dfa40104f2a1a2ec3fcfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.297ex; height:2.843ex;" alt="{\displaystyle P(z;c_{0},\delta ,M)}" loading="lazy"></span>, falls die Dirichlet-Reihe

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(s;z):=F(s)\zeta (s)^{-z}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/648923bf8767f9312ff3922db87291010fe00134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.509ex; height:3.009ex;" alt="{\displaystyle G(s;z):=F(s)\zeta (s)^{-z}}" loading="lazy"></span></dd></dl>

<p>eine auf dem ganzen <a href="Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\operatorname {Re} (s)>1-{\tfrac {c_{0}}{1+\max\{0,\log(|\operatorname {Im} (s)|)\}}}\right\}}">

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{\operatorname {Re} (s)>1-{\tfrac {c_{0}}{1+\max\{0,\log(|\operatorname {Im} (s)|)\}}}\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3508dd4ecd19338d4080dcb79b51ad6299c8948f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.459ex; height:4.843ex;" alt="{\displaystyle \left\{\operatorname {Re} (s)>1-{\tfrac {c_{0}}{1+\max\{0,\log(|\operatorname {Im} (s)|)\}}}\right\}}" loading="lazy"></span> holomorphe Funktion darstellt und dort außerdem der Ungleichung

</p>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G(s;z)|\leq M(1+|\operatorname {Im} (s)|)^{1-\delta }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38834c056c22f0a5f64d1a013db2d4b048944d73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.925ex; height:3.176ex;" alt="{\displaystyle |G(s;z)|\leq M(1+|\operatorname {Im} (s)|)^{1-\delta }}" loading="lazy"></span></dd></dl>

<p>genügt. Hierbei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbd45922057e4d7a5718ce5ed703ab493c63897a.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 \zeta (s)}" loading="lazy"></span> die <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannsche Zeta-Funktion</a>. Existiert nun eine Folge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(n)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45acf3d5914aa3ca6d1e328a7296f0c1ee805094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.202ex; height:2.843ex;" alt="{\displaystyle b(n)}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a(n)|\leq b(n)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a(n)|\leq b(n)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d719fc9def9e25528a1db9a0bedeb892b282371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.028ex; height:2.843ex;" alt="{\displaystyle |a(n)|\leq b(n)}" loading="lazy"></span> und die Reihe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum b(n)/n^{s}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum b(n)/n^{s}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3832a07dcae1c9517fc2c595cce49f1e798306d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle \textstyle \sum b(n)/n^{s}}" loading="lazy"></span> gehört zur Klasse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(w;c_{0},\delta ,M)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(w;c_{0},\delta ,M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1994ea0d5000eafd68166aaafb571a6ad95da94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.873ex; height:2.843ex;" alt="{\displaystyle P(w;c_{0},\delta ,M)}" loading="lazy"></span>, so liegt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d633e75b0c0edb9b5cf174df6f79f4b90634718b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.641ex; height:2.843ex;" alt="{\displaystyle F(s)}" loading="lazy"></span> per Definition sogar in der Klasse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z,w;c_{0},\delta ,M).}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(z,w;c_{0},\delta ,M).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2733573fb67e2b92e679ac45fe02010a0d6ca7c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.532ex; height:2.843ex;" alt="{\displaystyle T(z,w;c_{0},\delta ,M).}" loading="lazy"></span> Dann lässt sich die folgende modifizierte Funktion im Ursprung lokal als <a href="Taylor-Reihe" class="mw-redirect" title="Taylor-Reihe">Taylor-Reihe</a> schreiben:

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {s^{z}F(s+1)}{s+1}}=\sum _{k=0}^{\infty }\mu _{k}(z)s^{k},\quad |s|<\min\{c_{0},1\}.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/144b182b944e7c9fec16b66b68d136b9b7e4d279.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.923ex; height:7.009ex;" alt="{\displaystyle {\frac {s^{z}F(s+1)}{s+1}}=\sum _{k=0}^{\infty }\mu _{k}(z)s^{k},\quad |s|<\min\{c_{0},1\}.}" loading="lazy"></span></dd></dl>

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

<p>Liegt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d633e75b0c0edb9b5cf174df6f79f4b90634718b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.641ex; height:2.843ex;" alt="{\displaystyle F(s)}" loading="lazy"></span> in der Klasse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z,w;c_{0},\delta ,M)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(z,w;c_{0},\delta ,M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/285c5ad5d7e85ce0974da039e27822950c2a192a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.886ex; height:2.843ex;" alt="{\displaystyle T(z,w;c_{0},\delta ,M)}" loading="lazy"></span>, so gilt bereits für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 3,N\geq 0,A>0,|z|,|w|\leq A}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\geq 3,N\geq 0,A>0,|z|,|w|\leq A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e2d376df1f93b90c11beae680e71699a6b00f22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.236ex; height:2.843ex;" alt="{\displaystyle x\geq 3,N\geq 0,A>0,|z|,|w|\leq A}" loading="lazy"></span>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\leq x}a(n)=x(\log x)^{z-1}\left(\sum _{0\leq k\leq N}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O\left(M\left(\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}}+\left({\frac {c_{2}N+1}{\log(x)}}\right)^{N+1}\right)\right)\right).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cf1008b1e316869c8be55a21f787bd93bccbab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:95.896ex; height:7.676ex;" alt="{\displaystyle \sum _{n\leq x}a(n)=x(\log x)^{z-1}\left(\sum _{0\leq k\leq N}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O\left(M\left(\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}}+\left({\frac {c_{2}N+1}{\log(x)}}\right)^{N+1}\right)\right)\right).}" loading="lazy"></span></dd></dl>

<p>Hierbei hängen die positiven Konstanten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},c_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15ae7f7452e416f7b70940ca260b2582f10382cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.156ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2}}" loading="lazy"></span> und die implizite Konstante im <a href="Landau-Symbol" class="mw-redirect" title="Landau-Symbol">Landau-Symbol</a> höchstens von der Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0},\delta }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/184c04006a47f7b694720084b81f51314205d53e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.144ex; height:2.676ex;" alt="{\displaystyle c_{0},\delta }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ab. Ein wichtiger Spezialfall ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89579fe9cb330174ba72262b85764c9bcd827b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.479ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {Z} }" loading="lazy"></span>. Dann folgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\Gamma (z-k)=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\Gamma (z-k)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc144905e03d8c85cbfeb59ae7d69575dd5ed19a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.988ex; height:2.843ex;" alt="{\displaystyle 1/\Gamma (z-k)=0}" loading="lazy"></span>, wann immer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\geq z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b39ee9dd2faa3f1bc3167a79f6edbc191f32fd70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle k\geq z}" loading="lazy"></span> gilt. Dies ermöglicht es, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> so zu wählen, dass der Fehlerterm minimiert wird. Etwa erreicht man mit der Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N:=\lfloor \log(x)/\mathrm {e} c_{2}\rfloor }">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N:=\lfloor \log(x)/\mathrm {e} c_{2}\rfloor }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b78460ad3d3eabd357d46486fba61bb9690e123.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.24ex; height:2.843ex;" alt="{\displaystyle N:=\lfloor \log(x)/\mathrm {e} c_{2}\rfloor }" loading="lazy"></span> die Aussage

</p>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\leq x}a(n)=x(\log(x))^{z-1}\left(\sum _{0\leq k\leq z-1}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O\left(M\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}}\right)\right).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94240316463b211748670f560e1ba90195a153d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.737ex; height:7.676ex;" alt="{\displaystyle \sum _{n\leq x}a(n)=x(\log(x))^{z-1}\left(\sum _{0\leq k\leq z-1}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O\left(M\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}}\right)\right).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Vor-_und_Nachteile">Vor- und Nachteile</h2></div>

<p>Vorteile der Selberg-Delange-Methode sind die recht explizite Angabe eines Fehlerterms sowie die fehlende Notwendigkeit, dass die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/943e4e4e0ed21960105be2f6977215ed57e930d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.434ex; height:2.843ex;" alt="{\displaystyle a(n)}" loading="lazy"></span> stets nicht-negativ sein müssen. Jedoch kann die geforderte vertikale Abschätzung (die nicht weggelassen werden kann!) eine Hürde darstellen. Werden also weniger detaillierte Angaben über die mittlere Ordnung gebraucht, kann man auch auf <a href="Taubersatz" class="mw-redirect mw-disambig" title="Taubersatz">Taubersätze</a> zurückgreifen, die bereits unter deutlich schwächeren Annahmen gelten, jedoch keine Abschätzung der Fehlerterme zulassen.

</p>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="G%C3%A9rald_Tenenbaum" title="Gérald Tenenbaum">Gérald Tenenbaum</a>: <cite style="font-style:italic">Introduction to analytic and probabilistic number theory</cite>. AMS, Rhode Island 1990, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>281</span>.<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:Selberg-Delange-Methode&rft.au=G%C3%A9rald+Tenenbaum&rft.btitle=Introduction+to+analytic+and+probabilistic+number+theory&rft.date=1990&rft.genre=book&rft.pages=281&rft.place=Rhode+Island&rft.pub=AMS" style="display:none"> </span></span>

</li>

</ol></div></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-04-27" href="https://de.wikipedia.org/wiki/?title=Selberg-Delange-Methode&oldid=233205943">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>