GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Teilersumme</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Teilersumme</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">Unter der Teilersumme einer <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahl</a> versteht man die Summe aller positiven <a href="Teilbarkeit" title="Teilbarkeit">Teiler</a> dieser Zahl einschließlich der Zahl selbst.<a href="#cite_note-:0-1">[1]</a>

Beispiel:

<dl><dd>Die Zahl 6 hat die Teiler 1, 2, 3 und 6. Die Teilersumme von 6 lautet also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2+3+6=12}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+2+3+6=12}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46351b2f6f00f6672ead3274981505183fbfac1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.594ex; height:2.343ex;" alt="{\displaystyle 1+2+3+6=12}" loading="lazy">.</dd></dl>

Bei vielen Problemstellungen der Zahlentheorie spielen Teilersummen eine Rolle,

z. B. bei den <a href="Vollkommene_Zahl" title="Vollkommene Zahl">vollkommenen Zahlen</a> und den <a href="Befreundete_Zahlen" title="Befreundete Zahlen">befreundeten Zahlen</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Definition_1:_Summe_aller_Teiler">Definition 1: Summe aller Teiler</h3></div>

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},t_{2},...,t_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6da5224c02e5b890f300db7a1fa0b99a47fcec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.92ex; height:2.343ex;" alt="{\displaystyle t_{1},t_{2},...,t_{k}}" loading="lazy"> alle Teiler der natürlichen Zahl <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">, so nennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)=t_{1}+t_{2}+\dotsb +t_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)=t_{1}+t_{2}+\dotsb +t_{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d564bd06cbf1af2fc00c6e27e6976e79fabdb808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.593ex; height:2.843ex;" alt="{\displaystyle \sigma (n)=t_{1}+t_{2}+\dotsb +t_{k}}" loading="lazy"> die Teilersumme von <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">. Dabei sind 1 und <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"> selbst Teiler, also in der Menge der Teiler enthalten. Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mapsto \sigma (n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1697d9306cbd8efe2cc5ffcdb2c905da6af1704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.542ex; height:2.843ex;" alt="{\displaystyle n\mapsto \sigma (n)}" loading="lazy"> heißt Teilersummenfunktion und ist eine <a href="Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion">zahlentheoretische Funktion</a>.

Das Beispiel oben kann man nun so schreiben:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (6)=1+2+3+6=12}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d736df535e421abe0fa1859c19c52a064f746d7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.994ex; height:2.843ex;" alt="{\displaystyle \sigma (6)=1+2+3+6=12}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Definition_2:_Summe_der_echten_Teiler">Definition 2: Summe der echten Teiler</h3></div>

Die Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(n)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91043a6e32c30ce05e082c7fa67b2af06be57fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.589ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(n)}" loading="lazy"> der echten Teiler der natürlichen Zahl <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"> ist die Summe der Teiler von <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"> ohne die Zahl <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"> selbst.

Beispiel:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(6)=1+2+3=6}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cc901b163c8b92c8bfaf7cc41168cee1d55f7e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.884ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(6)=1+2+3=6}" loading="lazy">.</dd></dl>

Es gilt die Beziehung

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/750820c398578480347463194e57f28a8ab2a13b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.456ex; height:2.843ex;" alt="{\displaystyle \sigma (n)-n=\sigma ^{*}(n)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Definition_3:_defizient,_abundant,_vollkommen">Definition 3: defizient, abundant, vollkommen</h3></div>

Eine natürliche Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" loading="lazy"> heißt

<dl><dd><a href="Defiziente_Zahl" title="Defiziente Zahl">defizient</a> oder teilerarm, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(n)<n}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee2fafffe1afba2cd25b69de1b0ff6809c10e1af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.082ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(n)<n}" loading="lazy">,</dd>

<dd><a href="Abundante_Zahl" title="Abundante Zahl">abundant</a> oder teilerreich, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(n)>n}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a668884815fdcffdff0c5d05d3abd08d8989807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.082ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(n)>n}" loading="lazy">,</dd>

<dd><a href="Vollkommene_Zahl" title="Vollkommene Zahl">vollkommen</a>, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(n)=n}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a53d07cb577b4044ea76feabb3fd4bfbe6c4713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.082ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(n)=n}" loading="lazy">.<a href="#cite_note-2">[2]</a></dd></dl>

Beispiele:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(6)=1+2+3=6}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cc901b163c8b92c8bfaf7cc41168cee1d55f7e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.884ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(6)=1+2+3=6}" loading="lazy">, d. h. 6 ist eine vollkommene Zahl.</dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(12)=1+2+3+4+6=16>12}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf4df7f740acb9d3a7c55876feae5e1988611a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.638ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(12)=1+2+3+4+6=16>12}" loading="lazy">, d. h. 12 ist abundant.</dd>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(10)=1+2+5=8<10}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/842431f1b3fbdd15ce0f0a73f955a37bbaaf91f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.47ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(10)=1+2+5=8<10}" loading="lazy">, d. h. 10 ist defizient.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_der_Teilersumme">Eigenschaften der Teilersumme</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Satz_1:_Teilersumme_einer_Primzahl">Satz 1: Teilersumme einer Primzahl</h3></div>

Für jede <a href="Primzahl" title="Primzahl">Primzahl</a> <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"> gilt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (p)=p+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2268dccfdd339154032087e889425e9dce520858.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.579ex; height:2.843ex;" alt="{\displaystyle \sigma (p)=p+1}" loading="lazy">.</dd></dl>

Beweis: Per Definition hat <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"> nur die Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> und <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">. Daraus folgt die Behauptung.

<div class="mw-heading mw-heading3"><h3 id="Satz_2:_Teilersumme_der_Potenz_einer_Primzahl">Satz 2: Teilersumme der Potenz einer Primzahl</h3></div>

Sei <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"> eine Primzahl und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97bceb13f72e37bcd50b60e5fb2fa05bcf15c265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.784ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {N} _{0}}" loading="lazy">. Dann gilt für die <a href="Potenz_(Mathematik)" title="Potenz (Mathematik)">Potenz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e017c102135ab13bdf501dc1c1b5fd1840a97822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:3.009ex;" alt="{\displaystyle p^{k}}" loading="lazy">:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (p^{k})=\sum _{j=0}^{k}p^{j}={\frac {p^{k+1}-1}{p-1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f9913c14a1e069e9d9c4975cd11207464dc071b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.613ex; height:7.676ex;" alt="{\displaystyle \sigma (p^{k})=\sum _{j=0}^{k}p^{j}={\frac {p^{k+1}-1}{p-1}}}" loading="lazy">.</dd></dl>

Beweis: Da <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"> eine Primzahl ist, hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e017c102135ab13bdf501dc1c1b5fd1840a97822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:3.009ex;" alt="{\displaystyle p^{k}}" loading="lazy"> nur die Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{0},p^{1},\ldots ,p^{k}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{0},p^{1},\ldots ,p^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8abf17bff9f328e31af4007138a096ea02c14f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.007ex; height:3.009ex;" alt="{\displaystyle p^{0},p^{1},\ldots ,p^{k}}" loading="lazy">. Diese Zahlen bilden eine <a href="Geometrische_Reihe" title="Geometrische Reihe">geometrische Folge</a>. Aus der Formel für die Partialsummen der geometrischen Reihe folgt sofort die Behauptung.

Beispiel:

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (2^{3})={{2^{4}-1} \over {2-1}}={{16-1} \over 1}=15}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5af5aa031c5c61b979f1198aab1e8dafb4ff224f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.196ex; height:5.843ex;" alt="{\displaystyle \sigma (2^{3})={{2^{4}-1} \over {2-1}}={{16-1} \over 1}=15}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (8)=1+2+4+8=15}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18b5462e2e0cd84d29fe224d6a37bd6275ecdcbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.994ex; height:2.843ex;" alt="{\displaystyle \sigma (8)=1+2+4+8=15}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Satz_3:_Teilersumme_des_Produktes_von_zwei_Primzahlen">Satz 3: Teilersumme des Produktes von zwei Primzahlen</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> und <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"> verschiedene Primzahlen. Dann gilt

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54758a2c2aa00b857649aae6215b0a9854f9d1a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.328ex; height:2.843ex;" alt="{\displaystyle \sigma (a\cdot b)=\sigma (a)\cdot \sigma (b)}" loading="lazy">.</dd></dl>

Beweis: Die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49337c5cf256196e2292f7047cb5da68c24ca95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}" loading="lazy"> besitzt genau die Teiler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,a,b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e52c09ed73a1f87dda8f1c870cdaeba3499f61e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.458ex; height:2.509ex;" alt="{\displaystyle 1,a,b}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49337c5cf256196e2292f7047cb5da68c24ca95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}" loading="lazy">. Daraus folgt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (a\cdot b)=1+a+b+ab=(a+1)(b+1)=\sigma (a)\cdot \sigma (b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f882715207595b832b5533abf0ca8743d39a5e93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.515ex; height:2.843ex;" alt="{\displaystyle \sigma (a\cdot b)=1+a+b+ab=(a+1)(b+1)=\sigma (a)\cdot \sigma (b)}" loading="lazy">.</dd></dl>

Beispiel:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (3\cdot 5)=\sigma (15)=1+3+5+15=24}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84158b3ac31f3580fdb2cd9ad5cfcdf72cad9067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.561ex; height:2.843ex;" alt="{\displaystyle \sigma (3\cdot 5)=\sigma (15)=1+3+5+15=24}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (3)\cdot \sigma (5)=(1+3)\cdot (1+5)=4\cdot 6=24}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c381067bd3b9a2cd1fdde48bb22e9ec6ee23f1af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.535ex; height:2.843ex;" alt="{\displaystyle \sigma (3)\cdot \sigma (5)=(1+3)\cdot (1+5)=4\cdot 6=24}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Satz_4:_Teilersumme_des_Produkts_von_zwei_teilerfremden_Zahlen">Satz 4: Teilersumme des Produkts von zwei teilerfremden Zahlen</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> und <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"> teilerfremde Zahlen, so gilt

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54758a2c2aa00b857649aae6215b0a9854f9d1a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.328ex; height:2.843ex;" alt="{\displaystyle \sigma (a\cdot b)=\sigma (a)\cdot \sigma (b)}" loading="lazy">.<a href="#cite_note-3">[3]</a></dd></dl>

Die Teilersummenfunktion ist also <a href="Zahlentheoretische_Funktion#Multiplikative_Funktionen" title="Zahlentheoretische Funktion">multiplikativ</a>.

Beispiel:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (4\cdot 9)=\sigma (36)=1+2+3+4+6+9+12+18+36=91}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9980f64ab4673e49adb80e71761c550cbd0ba0da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.9ex; height:2.843ex;" alt="{\displaystyle \sigma (4\cdot 9)=\sigma (36)=1+2+3+4+6+9+12+18+36=91}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (4)\cdot \sigma (9)=(1+2+4)\cdot (1+3+9)=7\cdot 13=91}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08d88583e069621ee644b723a7a09db3d128d890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.703ex; height:2.843ex;" alt="{\displaystyle \sigma (4)\cdot \sigma (9)=(1+2+4)\cdot (1+3+9)=7\cdot 13=91}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Satz_5:_Teilersumme_einer_in_Primfaktoren_zerlegten_Zahl">Satz 5: Teilersumme einer in Primfaktoren zerlegten Zahl</h3></div>

Sei <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"> mit der <a href="Primfaktorzerlegung" title="Primfaktorzerlegung">Primfaktorzerlegung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p_{1}^{k_{1}}\cdot p_{2}^{k_{2}}\cdot \ldots \cdot p_{r}^{k_{r}}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=p_{1}^{k_{1}}\cdot p_{2}^{k_{2}}\cdot \ldots \cdot p_{r}^{k_{r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42d57bf8dd5cffacc1610af1665939b094874626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.458ex; height:3.509ex;" alt="{\displaystyle n=p_{1}^{k_{1}}\cdot p_{2}^{k_{2}}\cdot \ldots \cdot p_{r}^{k_{r}}}" loading="lazy">. Dann gilt

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)={\frac {p_{1}^{k_{1}+1}-1}{p_{1}-1}}\cdot \ldots \cdot {\frac {p_{r}^{k_{r}+1}-1}{p_{r}-1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc322d06dfb263418d699217d2639f68f863a4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.707ex; height:6.843ex;" alt="{\displaystyle \sigma (n)={\frac {p_{1}^{k_{1}+1}-1}{p_{1}-1}}\cdot \ldots \cdot {\frac {p_{r}^{k_{r}+1}-1}{p_{r}-1}}}" loading="lazy">.<a href="#cite_note-4">[4]</a></dd></dl>

Beispiel:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (84)=\sigma (2^{2}\cdot 3\cdot 7)={\frac {2^{3}-1}{2-1}}\cdot {\frac {3^{2}-1}{3-1}}\cdot {\frac {7^{2}-1}{7-1}}=7\cdot 4\cdot 8=224.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f177e11a1f8b11579827860816bd6744045973b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:64.402ex; height:5.843ex;" alt="{\displaystyle \sigma (84)=\sigma (2^{2}\cdot 3\cdot 7)={\frac {2^{3}-1}{2-1}}\cdot {\frac {3^{2}-1}{3-1}}\cdot {\frac {7^{2}-1}{7-1}}=7\cdot 4\cdot 8=224.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Satz_von_Thabit">Satz von Thabit</h2></div>

Mit Hilfe von Satz 4 kann man den Satz von Thabit (benannt nach <a href="Thabit_ibn_Qurra" title="Thabit ibn Qurra">Thabit ibn Qurra</a>) aus dem Gebiet der befreundeten Zahlen beweisen. Der Satz lautet:

Für eine natürliche Zahl <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"> seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=3\cdot 2^{n}-1,y=3\cdot 2^{n-1}-1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=3\cdot 2^{n}-1,y=3\cdot 2^{n-1}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1d362be11883fd7f72759ec6d1c8276ca7c41c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.267ex; height:3.009ex;" alt="{\displaystyle x=3\cdot 2^{n}-1,y=3\cdot 2^{n-1}-1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=9\cdot 2^{2n-1}-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=9\cdot 2^{2n-1}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e74d3c902d8c0c0aa454484fb18731c3d9aa37f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.334ex; height:2.843ex;" alt="{\displaystyle z=9\cdot 2^{2n-1}-1}" loading="lazy">.

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> Primzahlen größer als 2 sind, dann sind die beiden Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2^{n}xy}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=2^{n}xy}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae4e2830ab267315ac0d62e84556098c27a0d2ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.194ex; height:2.676ex;" alt="{\displaystyle a=2^{n}xy}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2^{n}z}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2^{n}z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3017bd397c3ac76934fc3f9814f0b3855c1f713d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.565ex; height:2.343ex;" alt="{\displaystyle b=2^{n}z}" loading="lazy"> befreundet, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(a)=b}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(a)=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6c36075f1a95716db650565a7fe51a498c74cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.52ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(a)=b}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(b)=a}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(b)=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baee5cf51a46bc12347a61bfe26b08c3fb64a23f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.52ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(b)=a}" loading="lazy">.

<dl><dt>Beweis</dt>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma ^{*}(a)&=\sigma (a)-a\\&=\sigma (2^{n}\cdot x\cdot y)-a\\&=(2^{n+1}-1)(x+1)(y+1)-a\qquad \qquad \qquad \qquad \qquad \qquad &&{\text{(Satz 4)}}\\&=(2^{n+1}-1)(3\cdot 2^{n})(3\cdot 2^{n-1})-2^{n}(3\cdot 2^{n}-1)(3\cdot 2^{n-1}-1)\\&=(2^{n+1}-1)\cdot 9\cdot 2^{2n-1}-2^{n}(9\cdot 2^{2n-1}-6\cdot 2^{n-1}-3\cdot 2^{n-1}+1)\\&=2\cdot 2^{n}\cdot 9\cdot 2^{2n-1}-9\cdot 2^{n}\cdot 2^{n-1}-2^{n}(9\cdot 2^{2n-1}-9\cdot 2^{n-1}+1)\\&=2^{n}(18\cdot 2^{2n-1}-9\cdot 2^{n-1}-9\cdot 2^{2n-1}+9\cdot 2^{n-1}-1)\\&=2^{n}(9\cdot 2^{2n-1}-1)\\&=2^{n}\cdot z\\&=b\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8dd0ba4046e253c6b2d91ffd051f917b6103a1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.505ex; width:81.486ex; height:32.176ex;" alt="{\displaystyle {\begin{aligned}\sigma ^{*}(a)&=\sigma (a)-a\\&=\sigma (2^{n}\cdot x\cdot y)-a\\&=(2^{n+1}-1)(x+1)(y+1)-a\qquad \qquad \qquad \qquad \qquad \qquad &&{\text{(Satz 4)}}\\&=(2^{n+1}-1)(3\cdot 2^{n})(3\cdot 2^{n-1})-2^{n}(3\cdot 2^{n}-1)(3\cdot 2^{n-1}-1)\\&=(2^{n+1}-1)\cdot 9\cdot 2^{2n-1}-2^{n}(9\cdot 2^{2n-1}-6\cdot 2^{n-1}-3\cdot 2^{n-1}+1)\\&=2\cdot 2^{n}\cdot 9\cdot 2^{2n-1}-9\cdot 2^{n}\cdot 2^{n-1}-2^{n}(9\cdot 2^{2n-1}-9\cdot 2^{n-1}+1)\\&=2^{n}(18\cdot 2^{2n-1}-9\cdot 2^{n-1}-9\cdot 2^{2n-1}+9\cdot 2^{n-1}-1)\\&=2^{n}(9\cdot 2^{2n-1}-1)\\&=2^{n}\cdot z\\&=b\end{aligned}}}" loading="lazy"></dd></dl>

Analog zeigt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{*}(b)=a}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ^{*}(b)=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baee5cf51a46bc12347a61bfe26b08c3fb64a23f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.52ex; height:2.843ex;" alt="{\displaystyle \sigma ^{*}(b)=a}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Teilersumme_als_endliche_Reihe">Teilersumme als endliche Reihe</h2></div>

Für jede natürliche Zahl <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"> kann die Teilerfunktion als Reihe dargestellt werden, ohne dass auf die Teilbarkeitseigenschaften

von <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"> explizit Bezug genommen wird:

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)=\sum _{\mu =1}^{n}\sum _{\nu =1}^{\mu }\cos {2\pi {\frac {\nu n}{\mu }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/060bd4e739ef88789d05ecf88a7fb15145b51832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.572ex; height:7.343ex;" alt="{\displaystyle \sigma (n)=\sum _{\mu =1}^{n}\sum _{\nu =1}^{\mu }\cos {2\pi {\frac {\nu n}{\mu }}}}" loading="lazy"></dd></dl>

Beweis:

Die Funktion

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(n,\mu )={\frac {1}{\mu }}\sum _{\nu =1}^{\mu }\cos 2\pi {\frac {\nu n}{\mu }},\quad n=1,2,\dots ,\quad \mu =1,2,\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/326b9777f033b275b026329ab27c2dd0165f3ded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.523ex; height:7.009ex;" alt="{\displaystyle T(n,\mu )={\frac {1}{\mu }}\sum _{\nu =1}^{\mu }\cos 2\pi {\frac {\nu n}{\mu }},\quad n=1,2,\dots ,\quad \mu =1,2,\dots }" loading="lazy"></dd></dl>

wird 1, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> ein Teiler von <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"> ist, ansonsten bleibt sie Null.

Sei nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> ein Teiler von <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">. Dann ist der Quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nu n}{\mu }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4310536da010c917794d79fc815020c0432b190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.463ex; height:5.176ex;" alt="{\displaystyle {\frac {\nu n}{\mu }}}" loading="lazy"> ganzzahlig, somit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {2\pi {\frac {\nu n}{\mu }}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {2\pi {\frac {\nu n}{\mu }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d294dd2c63d3cfd86a93ce0ba8ef9fd13dd282a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.456ex; height:5.176ex;" alt="{\displaystyle \cos {2\pi {\frac {\nu n}{\mu }}}}" loading="lazy"> gleich 1. Die Summation über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> ergibt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">, woraus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n,\mu )=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13c90635f2119444ac16dd10548bd2a9ab508040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.537ex; height:2.843ex;" alt="{\displaystyle T(n,\mu )=1}" loading="lazy"> folgt.

Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> kein Teiler von <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">.

Es gilt dann

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(n,\mu )={\frac {1}{\mu }}\sum _{\nu =1}^{\mu }\cos 2\pi {\frac {\nu n}{\mu }}={\frac {1}{\mu }}{\frac {\sin \pi n\cos {\frac {\pi (\mu +1)n}{\mu }}}{\sin {\frac {\pi n}{\mu }}}}=0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa04def893c001c70cd0d8efe103bf2bdeabc33c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.267ex; height:8.509ex;" alt="{\displaystyle T(n,\mu )={\frac {1}{\mu }}\sum _{\nu =1}^{\mu }\cos 2\pi {\frac {\nu n}{\mu }}={\frac {1}{\mu }}{\frac {\sin \pi n\cos {\frac {\pi (\mu +1)n}{\mu }}}{\sin {\frac {\pi n}{\mu }}}}=0.}" loading="lazy"></dd></dl>

Damit ist gezeigt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n,\mu )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2902cd7c5ab8d9521f7f324db89806c374ac347d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.276ex; height:2.843ex;" alt="{\displaystyle T(n,\mu )}" loading="lazy"> genau dann gleich 1 ist, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> ein Teiler von <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"> ist, und ansonsten verschwindet.

Multipliziert man jetzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(n,\mu )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2902cd7c5ab8d9521f7f324db89806c374ac347d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.276ex; height:2.843ex;" alt="{\displaystyle T(n,\mu )}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79f768f87ee222ea1c3522e8a793f40c288928eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:3.176ex;" alt="{\displaystyle \mu ^{k}}" loading="lazy"> und summiert das Produkt über alle Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/820c85551af65c6bedaf1b895fbd99bd9e23ec4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =1}" loading="lazy"> bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a93a9aeaca479534704eaadde6561f52863d283c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.895ex; height:2.176ex;" alt="{\displaystyle \mu =n}" loading="lazy">, so entsteht nur dann ein Beitrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79f768f87ee222ea1c3522e8a793f40c288928eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:3.176ex;" alt="{\displaystyle \mu ^{k}}" loading="lazy"> zur Summe, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> ein Teiler von <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"> ist. Das ist aber genau die Definition der allgemeinen <a href="Teilerfunktion" title="Teilerfunktion">Teilerfunktion</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}(n)=\sum _{\mu =1}^{n}\mu ^{k-1}\sum _{\nu =1}^{\mu }\cos {2\pi {\frac {\nu n}{\mu }}},\quad k=0,\pm 1,\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0e0455460e9a116d881d21e79fc7843d3d92fdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.227ex; height:7.343ex;" alt="{\displaystyle \sigma _{k}(n)=\sum _{\mu =1}^{n}\mu ^{k-1}\sum _{\nu =1}^{\mu }\cos {2\pi {\frac {\nu n}{\mu }}},\quad k=0,\pm 1,\dots }" loading="lazy"></dd></dl>

deren Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy"> die einfache Teilersumme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3213bbd075ed40d41d2b72dfaa4a190e031d3fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle \sigma (n)}" loading="lazy"> ist.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Inhaltskette" title="Inhaltskette">Inhaltskette</a></li>

<li><a href="Teilermenge" title="Teilermenge">Teilermenge</a></li></ul>

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

<ul><li><a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a>, <a href="J%C3%A1nos_Sur%C3%A1nyi" title="János Surányi">János Surányi</a>: <cite style="font-style:italic">Topics in the Theory of Numbers.</cite> (= <cite style="font-style:italic">Undergraduate Texts in Mathematics</cite>). 2. Auflage. Springer Verlag, New York, NY (u. a.) 2003, ISBN 0-387-95320-5 (Aus dem Ungarischen übersetzt von Barry Guiduli).<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:Teilersumme&rft.au=Paul+Erd%C5%91s%2C+J%C3%A1nos+Sur%C3%A1nyi&rft.btitle=Topics+in+the+Theory+of+Numbers.&rft.date=2003&rft.edition=2.&rft.genre=book&rft.isbn=0387953205&rft.place=New+York%2C+NY+%28u.+a.%29&rft.pub=Springer+Verlag&rft.series=Undergraduate+Texts+in+Mathematics" style="display:none"> </li>

<li>József Sándor, Dragoslav S. Mitrinović, Borislav Crstici: <cite style="font-style:italic">Handbook of Number Theory. I</cite>. Springer Verlag, Dordrecht 2006, ISBN 1-4020-4215-9.<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:Teilersumme&rft.au=J%C3%B3zsef+S%C3%A1ndor%2C+Dragoslav+S.+Mitrinovi%C4%87%2C+Borislav+Crstici&rft.btitle=Handbook+of+Number+Theory.+I&rft.date=2006&rft.genre=book&rft.isbn=1402042159&rft.place=Dordrecht&rft.pub=Springer+Verlag" style="display:none"> </li>

<li>József Sándor, Borislav Crstici: <cite style="font-style:italic">Handbook of Number Theory. II</cite>. Kluwer Academic Publishers, Dordrecht/Boston/London 2004, ISBN 1-4020-2546-7.<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:Teilersumme&rft.au=J%C3%B3zsef+S%C3%A1ndor%2C+Borislav+Crstici&rft.btitle=Handbook+of+Number+Theory.+II&rft.date=2004&rft.genre=book&rft.isbn=1402025467&rft.place=Dordrecht%2FBoston%2FLondon&rft.pub=Kluwer+Academic+Publishers" style="display:none"> </li>

<li><a href="Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a>: <cite style="font-style:italic">Elementary Theory of Numbers</cite> (= <cite style="font-style:italic">North-Holland Mathematical Library</cite>. Band 31). 2. überarbeitete und erweiterte Auflage. North-Holland, Amsterdam / New York 1988, ISBN 0-444-86662-0.<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:Teilersumme&rft.au=Wac%C5%82aw+Sierpi%C5%84ski&rft.btitle=Elementary+Theory+of+Numbers&rft.date=1988&rft.edition=2.+%C3%BCberarbeitete+und+erweiterte&rft.genre=book&rft.isbn=0444866620&rft.place=Amsterdam+%2F+New+York&rft.pub=North-Holland&rft.series=North-Holland+Mathematical+Library" style="display:none"> </li>

<li><a href="Jochen_Ziegenbalg" title="Jochen Ziegenbalg">Jochen Ziegenbalg</a>: <cite style="font-style:italic">Elementare Zahlentheorie. Beispiele, Geschichte, Algorithmen</cite>. 2. Auflage. Springer, Wiesbaden 2015, ISBN 978-3-658-07170-7.<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:Teilersumme&rft.au=Jochen+Ziegenbalg&rft.btitle=Elementare+Zahlentheorie.+Beispiele%2C+Geschichte%2C+Algorithmen&rft.date=2015&rft.edition=2.&rft.genre=book&rft.isbn=9783658071707&rft.place=Wiesbaden&rft.pub=Springer" style="display:none"> </li></ul>

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

<li id="cite_note-:0-1"><a href="#cite_ref-:0_1-0">↑</a> Jochen Ziegenbalg: <cite style="font-style:italic">Elementare Zahlentheorie</cite>. 2. Auflage. 2015, S. 35.<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:Teilersumme&rft.au=Jochen+Ziegenbalg&rft.btitle=Elementare+Zahlentheorie&rft.date=2015&rft.edition=2.&rft.genre=book&rft.pages=35" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Jochen Ziegenbalg: <cite style="font-style:italic">Elementare Zahlentheorie</cite>. 2. Auflage. 2015, S. 37.<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:Teilersumme&rft.au=Jochen+Ziegenbalg&rft.btitle=Elementare+Zahlentheorie&rft.date=2015&rft.edition=2.&rft.genre=book&rft.pages=37" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Jochen Ziegenbalg: <cite style="font-style:italic">Elementare Zahlentheorie</cite>. 2. Auflage. 2015, S. 36.<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:Teilersumme&rft.au=Jochen+Ziegenbalg&rft.btitle=Elementare+Zahlentheorie&rft.date=2015&rft.edition=2.&rft.genre=book&rft.pages=36" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="G._H._Hardy" class="mw-redirect" title="G. H. Hardy">G. H. Hardy</a>, E. M. Wright: <cite style="font-style:italic">An Introduction to the Theory of Numbers</cite>. 4. Auflage. Oxford University Press, Oxford 1975, ISBN 0-19-853310-1, S. 239.<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:Teilersumme&rft.au=G.+H.+Hardy%2C+E.+M.+Wright&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.date=1975&rft.edition=4.&rft.genre=book&rft.isbn=0198533101&rft.pages=239&rft.place=Oxford&rft.pub=Oxford+University+Press" style="display:none">

</li>

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

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-12-08" href="https://de.wikipedia.org/wiki/?title=Teilersumme&oldid=262236875">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>