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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lambert-Reihe</span></h1>
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<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>In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist eine <b>Lambert-Reihe</b> eine spezielle <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a>. Benannt ist sie nach <a href="Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Johann Heinrich Lambert</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die Lambert-Reihe ist eine Reihe mit dieser Form:
</p>
<dl><dd><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 S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msup>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</mfrac>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee6616ab19deda72ea9a5156af7534a20b432a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.804ex; height:6.843ex;" alt="{\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}}" loading="lazy"></span></dd></dl>
<p>Die Lambertsche L-Funktion bilden den Spezialfall dieser Reihe 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}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle a_{n}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78a4b76527ad3c83b5ce1c7b42269408116e0728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.709ex; height:2.509ex;" alt="{\displaystyle a_{n}=1}" loading="lazy"></span> für alle Werte <i>n:</i>
</p>
<dl><dd><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 L(q)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msup>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</mfrac>
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<annotation encoding="application/x-tex">{\displaystyle L(q)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61e741620f33873634167752461c88e9c0d95946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.439ex; height:6.843ex;" alt="{\displaystyle L(q)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Konvergenz">Konvergenz</h3></div>
<p>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 |q|=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |q|=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5678803d701a5cf4019dfbeb476ec02c05ce7009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|=1}" loading="lazy"></span> konvergiert die Lambert-Reihe nicht. 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 |q|\neq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≠<!-- ≠ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |q|\neq 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efbd73b955732cc13e24edbeab2f869568bb493a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|\neq 1}" loading="lazy"></span> konvergiert sie stets dann, wenn 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 _{n=1}^{\infty }a_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64257f5068acc5587831589c5ec818722bf33a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}" loading="lazy"></span> konvergiert. Konvergiert <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 _{n=1}^{\infty }a_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64257f5068acc5587831589c5ec818722bf33a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}}" loading="lazy"></span> nicht, dann konvergiert die Lambert-Reihe für alle <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 q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>, für die die Potenzreihe <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 _{n=1}^{\infty }a_{n}q^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}q^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cfdc473ff362f2c345406d339f2f68dc38c135e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.906ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{n=1}^{\infty }a_{n}q^{n}}" loading="lazy"></span> konvergiert <i>(Satz von <a href="Konrad_Knopp" title="Konrad Knopp">Konrad Knopp</a>).</i>
</p>
<div class="mw-heading mw-heading3"><h3 id="Lambert-Reihe_als_Potenzreihe">Lambert-Reihe als Potenzreihe</h3></div>
<p>Die Lambert-Reihe kann 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 |q|<1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |q|&lt;1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef46eeba8cb2497a797ec6652c0f76d69e2175bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.624ex; height:2.843ex;" alt="{\displaystyle |q|<1}" loading="lazy"></span> in eine <a href="Geometrische_Reihe" title="Geometrische Reihe">geometrische Reihe</a>
</p>
<dl><dd><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 S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msub>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munderover>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mi>k</mi>
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</msup>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2bc4d620ad97bd3583b76855fdc807ba7af5f7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.406ex; height:6.843ex;" alt="{\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}}" loading="lazy"></span></dd></dl>
<p>entwickelt werden, wobei sich die Koeffizienten <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_{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/330e36691c8285b9c890d82a7c11855f3259dd58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.673ex; height:2.509ex;" alt="{\displaystyle b_{m}}" loading="lazy"></span> der neuen Reihe durch <a href="Dirichlet-Faltung" class="mw-redirect" title="Dirichlet-Faltung">Dirichlet-Faltung</a> 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 a_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> mit der konstanten 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 1_{(n)}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1_{(n)}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2081d026346a9b3a2b17290ff4f575632ab93267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.921ex; height:3.009ex;" alt="{\displaystyle 1_{(n)}=1}" loading="lazy"></span> ergeben:
</p>
<dl><dd><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_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>∗<!-- ∗ --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>∣<!-- ∣ --></mo>
<mi>m</mi>
</mrow>
</munder>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3decfa6aae9bbc62231b9e448dabf348296b02bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.306ex; height:6.009ex;" alt="{\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Alternative_Form">Alternative Form</h2></div>
<p>Setzt man <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 q=e^{-z}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>z</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q=e^{-z}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/245bb49f2d24b902dc46ef37941adf518093b8cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.532ex; height:2.843ex;" alt="{\displaystyle q=e^{-z}}" loading="lazy"></span>, so erhält man eine andere übliche Form der Reihe
</p>
<dl><dd><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 \sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>m</mi>
<mi>z</mi>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb96ea8a3de2139e6112d46be76b4227440acd39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.807ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz},}" loading="lazy"></span></dd></dl>
<p>wieder mit
</p>
<dl><dd><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_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>∗<!-- ∗ --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>∣<!-- ∣ --></mo>
<mi>m</mi>
</mrow>
</munder>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc73df0634cc567a4e98bbbef1bdb0a6b86dfe90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.953ex; height:6.009ex;" alt="{\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.}" loading="lazy"></span></dd></dl>
<p>Beispiele der Lambert-Reihe in dieser Form, 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 z=2\pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=2\pi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dec86f3f9d6a742888612104015d4a886ea33b43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.681ex; height:2.176ex;" alt="{\displaystyle z=2\pi }" loading="lazy"></span>, treten in Ausdrücken der <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannschen Zeta-Funktion</a> für ungerade natürliche Zahlen auf.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>
<p>Einige unendliche Summen können durch die Lambertsche L-Funktion dargestellt werden.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Unendliche Summe geradstelliger <a href="Fibonacci-Folge" title="Fibonacci-Folge">Fibonacci-Zahlen</a> (mit dem griechischen Buchstaben Phi wird die <a href="Goldener_Schnitt" title="Goldener Schnitt">Goldene Zahl</a> dargestellt):
</p>
<dl><dd><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 \sum _{n=1}^{\infty }{\frac {1}{F_{2n}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\ \Phi ^{2n}}{\Phi ^{4n}-1}}={\sqrt {5}}\left(L\left(\Phi ^{-2}\right)-L\left(\Phi ^{-4}\right)\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mtext>&nbsp;</mtext>
<msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>L</mi>
<mrow>
<mo>(</mo>
<msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mrow>
<mo>(</mo>
<msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\ \Phi ^{2n}}{\Phi ^{4n}-1}}={\sqrt {5}}\left(L\left(\Phi ^{-2}\right)-L\left(\Phi ^{-4}\right)\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d39a6be178bf7ddbbd9781a241f08369b944abe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.287ex; height:7.009ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n}}}=\sum _{n=1}^{\infty }{\frac {{\sqrt {5}}\ \Phi ^{2n}}{\Phi ^{4n}-1}}={\sqrt {5}}\left(L\left(\Phi ^{-2}\right)-L\left(\Phi ^{-4}\right)\right)}" loading="lazy"></span></dd></dl>
<p>Unendliche Summe der Kehrwerte geradstelliger <a href="Pell-Folge" title="Pell-Folge">Pell-Zahlen</a>:
</p>
<dl><dd><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 \sum _{n=1}^{\infty }{\frac {1}{P_{2n}}}=\sum _{n=1}^{\infty }{\frac {2{\sqrt {2}}\left({\sqrt {2}}+1\right)^{2n}}{\left({\sqrt {2}}+1\right)^{4n}-1}}=2{\sqrt {2}}\left(L\left(\left({\sqrt {2}}-1\right)^{2}\right)-L\left(\left({\sqrt {2}}-1\right)^{4}\right)\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>L</mi>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mi>L</mi>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{2n}}}=\sum _{n=1}^{\infty }{\frac {2{\sqrt {2}}\left({\sqrt {2}}+1\right)^{2n}}{\left({\sqrt {2}}+1\right)^{4n}-1}}=2{\sqrt {2}}\left(L\left(\left({\sqrt {2}}-1\right)^{2}\right)-L\left(\left({\sqrt {2}}-1\right)^{4}\right)\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcfa4a42aceaec137db6bf01d9a7e9e52fd89097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:74.743ex; height:8.509ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{2n}}}=\sum _{n=1}^{\infty }{\frac {2{\sqrt {2}}\left({\sqrt {2}}+1\right)^{2n}}{\left({\sqrt {2}}+1\right)^{4n}-1}}=2{\sqrt {2}}\left(L\left(\left({\sqrt {2}}-1\right)^{2}\right)-L\left(\left({\sqrt {2}}-1\right)^{4}\right)\right)}" loading="lazy"></span></dd></dl>
<p>Darstellung der <a href="Erd%C5%91s-Borwein-Konstante" title="Erdős-Borwein-Konstante">Erdős-Borwein-Konstante</a> mit der Lambertschen L-Funktion:
</p>
<dl><dd><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 E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}=\sum _{n=1}^{\infty }{\frac {2^{n}+1}{2^{n^{2}}\left(2^{n}-1\right)}}=L\left({\frac {1}{2}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<mrow>
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<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mn>1</mn>
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</mrow>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}=\sum _{n=1}^{\infty }{\frac {2^{n}+1}{2^{n^{2}}\left(2^{n}-1\right)}}=L\left({\frac {1}{2}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bf06a8399818d1265824dc8198d175009876d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.794ex; height:7.009ex;" alt="{\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}=\sum _{n=1}^{\infty }{\frac {2^{n}+1}{2^{n^{2}}\left(2^{n}-1\right)}}=L\left({\frac {1}{2}}\right)}" loading="lazy"></span></dd></dl>
<p>Unendliche Summe der Kehrwerte der Nachfolger der Zweierpotenzen:
</p>
<dl><dd><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 \sum _{n=1}^{\infty }{\frac {1}{2^{n}+1}}=\sum _{n=1}^{\infty }\left({\frac {1}{2^{n}-1}}-{\frac {2}{4^{n}-1}}\right)=L\left({\frac {1}{2}}\right)-2L\left({\frac {1}{4}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mrow>
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<mn>1</mn>
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<mn>1</mn>
<mn>4</mn>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}+1}}=\sum _{n=1}^{\infty }\left({\frac {1}{2^{n}-1}}-{\frac {2}{4^{n}-1}}\right)=L\left({\frac {1}{2}}\right)-2L\left({\frac {1}{4}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5110f847d03524c3537cbf3bca0aece302803941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.385ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}+1}}=\sum _{n=1}^{\infty }\left({\frac {1}{2^{n}-1}}-{\frac {2}{4^{n}-1}}\right)=L\left({\frac {1}{2}}\right)-2L\left({\frac {1}{4}}\right)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Thetafunktion" class="mw-redirect" title="Thetafunktion">Thetafunktion</a></li>
<li><a href="Elliptisches_Nomen" title="Elliptisches Nomen">Elliptisches Nomen</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LambertSeries.html"><i>Lambert Series</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a href="Gregor_Michailowitsch_Fichtenholz" title="Gregor Michailowitsch Fichtenholz">G. M. Fichtenholz</a>: <cite style="font-style:italic">Differential- und Integralrechnung II</cite> (=&nbsp;<cite style="font-style:italic">Hochschulbücher für Mathematik</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em">&nbsp;</span>62</span>). 6. Auflage. <a href="Volkseigener_Betrieb" title="Volkseigener Betrieb">VEB</a> <a href="Deutscher_Verlag_der_Wissenschaften" title="Deutscher Verlag der Wissenschaften">Deutscher Verlag der Wissenschaften</a>, Berlin 1974, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>323</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Lambert-Reihe&amp;rft.au=G.+M.+Fichtenholz&amp;rft.btitle=Differential-+und+Integralrechnung+II&amp;rft.date=1974&amp;rft.edition=6.&amp;rft.genre=book&amp;rft.pages=323&amp;rft.place=Berlin&amp;rft.pub=VEB+Deutscher+Verlag+der+Wissenschaften&amp;rft.series=Hochschulb%C3%BCcher+f%C3%BCr+Mathematik" style="display:none">&nbsp;</span></li>
<li>Ravi Agarwal: <i>Lambert series and Ramanujan.</i> Department of Mathematics and Astronomy, Universität Lucknow (लखनऊ विश्वविद्यालय), Indien.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="cite">Jonathan M. Borwein, Peter B. Borwein: <a rel="nofollow" class="external text" href="http://wayback.cecm.sfu.ca/~pborwein/TEMP_PROTECTED/pi-agm.pdf"><i>Pi and the AGM.</i></a> In: <i>wayback.cecm.sfu.ca.</i><span class="Abrufdatum"> Abgerufen am 12.&nbsp;Mai 2023</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&amp;rfr_id=info%3Asid%2Fde.wikipedia.org%3ALambert-Reihe&amp;rft.title=Pi+and+the+AGM&amp;rft.description=Pi+and+the+AGM&amp;rft.identifier=http%3A%2F%2Fwayback.cecm.sfu.ca%2F%7Epborwein%2FTEMP_PROTECTED%2Fpi-agm.pdf&amp;rft.creator=Jonathan+M.+Borwein%2C+Peter+B.+Borwein">&nbsp;</span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Erdos-BorweinConstant.html"><i>Erdős-Borwein Constant</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch). </span>
</li>
</ol></div><!--htdig_noindex--><div><div class="zim-footer">
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