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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Tschebyschow-Funktion</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>Die <b>Tschebyschow-Funktion</b>, etwa im Englischen auch <b>Chebyshev function</b> oder ähnlich bezeichnet, ist eine von zwei <a href="Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion">zahlentheoretischen Funktionen</a>, die nach dem russischen Mathematiker <a href="Pafnuti_Lwowitsch_Tschebyschow" title="Pafnuti Lwowitsch Tschebyschow">Pafnuti Lwowitsch Tschebyschow</a> benannt sind. Sie erhalten Bedeutung durch ihren Zusammenhang mit der Primzahlzählfunktion und dem <a href="Primzahlsatz" title="Primzahlsatz">Primzahlsatz</a> und damit der <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannschen Zeta-Funktion</a>.
</p><p>Die <b>erste Tschebyschow-Funktion</b>, üblicherweise 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 \theta \,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mspace width="thinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle \theta \,}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/228647b7d4a18b6c8c0c390b439a61da8fafec76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \theta \,}" loading="lazy"></span> oder <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 \vartheta }">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00eaf197c35bbfa391b9477490a4af955416837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }" loading="lazy"></span> bezeichnet, ist die Summe der <a href="Nat%C3%BCrlicher_Logarithmus" class="mw-redirect" title="Natürlicher Logarithmus">Logarithmen</a> der <a href="Primzahlen" class="mw-redirect" title="Primzahlen">Primzahlen</a> bis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
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<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
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</math></span><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"></span>:
</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 \vartheta (x)=\sum _{p\leq x \atop p{\text{ prim}}}\operatorname {log} (p)}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi>p</mi>
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<mtext>&nbsp;prim</mtext>
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<mi>log</mi>
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<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\sum _{p\leq x \atop p{\text{ prim}}}\operatorname {log} (p)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2ccec04490b780f990e937ce9efe048e4b04515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:18.506ex; height:7.176ex;" alt="{\displaystyle \vartheta (x)=\sum _{p\leq x \atop p{\text{ prim}}}\operatorname {log} (p)}" loading="lazy"></span></dd></dl>
<p>Die <b>zweite Tschebyschow-Funktion</b> <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 \psi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy"></span> ist die summierte Funktion der <a href="Mangoldt-Funktion" title="Mangoldt-Funktion">Mangoldt-Funktion</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 \psi (x)=\sum _{n=1}^{x}\Lambda (n)=\sum _{p^{k}\leq x}\operatorname {log} (p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
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<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>x</mi>
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</munderover>
<mi mathvariant="normal">Λ<!-- Λ --></mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
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</munder>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{n=1}^{x}\Lambda (n)=\sum _{p^{k}\leq x}\operatorname {log} (p)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85b71cda0bf7fe2ef87656af545e32c8a934aaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:29.651ex; height:7.676ex;" alt="{\displaystyle \psi (x)=\sum _{n=1}^{x}\Lambda (n)=\sum _{p^{k}\leq x}\operatorname {log} (p)}" loading="lazy"></span></dd></dl>
<p>wobei die Mangoldt-Funktion <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 \Lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Λ<!-- Λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"></span> definiert ist als
</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 \Lambda (n)={\begin{cases}\log(p)&amp;{\text{falls }}n{\text{ sich als }}n=p^{k}{\text{ darstellen lässt, wobei }}p{\text{ prim, }}k\in \mathbb {N} ^{+}\\0&amp;{\text{sonst}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Λ<!-- Λ --></mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>falls&nbsp;</mtext>
</mrow>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;sich als&nbsp;</mtext>
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<mi>n</mi>
<mo>=</mo>
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<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;darstellen lässt, wobei&nbsp;</mtext>
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<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;prim,&nbsp;</mtext>
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<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
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<mtd>
<mn>0</mn>
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<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sonst</mtext>
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<mo fence="true" stretchy="true" symmetric="true"></mo>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Lambda (n)={\begin{cases}\log(p)&amp;{\text{falls }}n{\text{ sich als }}n=p^{k}{\text{ darstellen lässt, wobei }}p{\text{ prim, }}k\in \mathbb {N} ^{+}\\0&amp;{\text{sonst}}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c590ddc903bff82df5cf8c1bc5a636b9c509d618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.276ex; height:6.176ex;" alt="{\displaystyle \Lambda (n)={\begin{cases}\log(p)&amp;{\text{falls }}n{\text{ sich als }}n=p^{k}{\text{ darstellen lässt, wobei }}p{\text{ prim, }}k\in \mathbb {N} ^{+}\\0&amp;{\text{sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Grundlegende_Eigenschaften">Grundlegende Eigenschaften</h2></div>
<p>Erstere Tschebyschow-Funktion lässt sich auch darstellen als
</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 \vartheta (x)=\log(x_{\#}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">#<!-- # --></mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\log(x_{\#}),}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d2fb58f84a0f71d4fa8b0928d27b5d3042daf37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.97ex; height:3.009ex;" alt="{\displaystyle \vartheta (x)=\log(x_{\#}),}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\#}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">#<!-- # --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{\#}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d1f04e88bf5e9857b5b0b1eb93b2fb983bef31a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.931ex; height:2.343ex;" alt="{\displaystyle x_{\#}}" loading="lazy"></span> die <a href="Primfakult%C3%A4t" class="mw-redirect" title="Primfakultät">Primfakultät</a> bezeichnet.
</p><p>Die zweite lässt sich auch schreiben als der Logarithmus des <a href="Kleinstes_gemeinsames_Vielfaches" title="Kleinstes gemeinsames Vielfaches">kleinsten gemeinsamen Vielfachen</a> von 1 bis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
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</math></span><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"></span>:
</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 \psi (x)=\operatorname {log} (\operatorname {kgV} (1,2,3,\ldots ,\lfloor x\rfloor ))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>kgV</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>x</mi>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\operatorname {log} (\operatorname {kgV} (1,2,3,\ldots ,\lfloor x\rfloor ))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0c2f3525b38025dd9129b963c54baac07b7b0db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.602ex; height:2.843ex;" alt="{\displaystyle \psi (x)=\operatorname {log} (\operatorname {kgV} (1,2,3,\ldots ,\lfloor x\rfloor ))}" loading="lazy"></span></dd></dl>
<p>Nach <a href="Erhard_Schmidt_(Mathematiker)" title="Erhard Schmidt (Mathematiker)">Erhard Schmidt</a> gibt es für jedes positive <a href="Reelle_Zahl" title="Reelle Zahl">reelle</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> Werte für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><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"></span>, sodass
</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 \psi (x)-x<-k{\sqrt {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>&lt;</mo>
<mo>−<!-- − --></mo>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)-x&lt;-k{\sqrt {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2be99086d1f80f54bdcd44a51e3efbf773dcbe8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.206ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x<-k{\sqrt {x}}}" loading="lazy"></span></dd></dl>
<p>und
</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 \psi (x)-x>k{\sqrt {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>&gt;</mo>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)-x&gt;k{\sqrt {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a776eb0c90174a15bc15626e45075478f5c8bfdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.397ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x>k{\sqrt {x}}}" loading="lazy"></span></dd></dl>
<p>unendlich oft.
</p>
<div class="mw-heading mw-heading3"><h3 id="Asymptotik">Asymptotik</h3></div>
<p>Es gilt
</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 \lim _{x\to \infty }{\frac {x}{\vartheta (x)}}=1,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mrow>
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }{\frac {x}{\vartheta (x)}}=1,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0986027c0b54b8833f6363b3bd3ddcf113542173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.87ex; height:5.509ex;" alt="{\displaystyle \lim _{x\to \infty }{\frac {x}{\vartheta (x)}}=1,}" loading="lazy"></span></dd></dl>
<p>d.&nbsp;h.
</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 \vartheta (n)\sim n.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>∼<!-- ∼ --></mo>
<mi>n</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vartheta (n)\sim n.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c17c53f79c723427bfd8210469768c4de9ac2b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.718ex; height:2.843ex;" alt="{\displaystyle \vartheta (n)\sim n.}" loading="lazy"></span></dd></dl>
<p>Ebenso gilt
</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 \psi (n)\sim n.\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>∼<!-- ∼ --></mo>
<mi>n</mi>
<mo>.</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (n)\sim n.\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29f90df817071017c7e1de7660d421a61d2f70ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle \psi (n)\sim n.\,}" loading="lazy"></span></dd></dl>
<p>Pierre Dusart fand eine Reihe von Schranken für die beiden Funktionen:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<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 \vartheta (p_{k})\geq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2{,}0553}{\ln k}}\right),\qquad k\geq \exp(22)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>≥<!-- ≥ --></mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>2,055</mn>
<mn>3</mn>
</mrow>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>k</mi>
<mo>≥<!-- ≥ --></mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>22</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vartheta (p_{k})\geq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2{,}0553}{\ln k}}\right),\qquad k\geq \exp(22)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eceebbd6c2714e812604378f0d5c70511c3078ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.707ex; height:6.176ex;" alt="{\displaystyle \vartheta (p_{k})\geq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2{,}0553}{\ln k}}\right),\qquad k\geq \exp(22)}" loading="lazy"></span></dd></dl>
<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 \vartheta (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right),\qquad k\geq 198}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>k</mi>
<mo>≥<!-- ≥ --></mo>
<mn>198</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vartheta (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right),\qquad k\geq 198}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24207a7f5a33b29406ea75cc4e22f1153c58f951.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.211ex; height:6.176ex;" alt="{\displaystyle \vartheta (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right),\qquad k\geq 198}" loading="lazy"></span></dd></dl>
<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 \psi (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right)+1{,}43{\sqrt {x}},\qquad k\geq 198}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>k</mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>43</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>k</mi>
<mo>≥<!-- ≥ --></mo>
<mn>198</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right)+1{,}43{\sqrt {x}},\qquad k\geq 198}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c68ff5c598a7185ea93fa6799a518a5447aed1b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.204ex; height:6.176ex;" alt="{\displaystyle \psi (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right)+1{,}43{\sqrt {x}},\qquad k\geq 198}" loading="lazy"></span></dd></dl>
<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 |\vartheta (x)-x|\leq 0{,}006788\,{\frac {x}{\ln x}},\qquad x\geq 10{.}544{.}111}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mn>0,006</mn>
<mn>788</mn>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>x</mi>
<mo>≥<!-- ≥ --></mo>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>.</mo>
</mrow>
<mn>544</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>.</mo>
</mrow>
<mn>111</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\vartheta (x)-x|\leq 0{,}006788\,{\frac {x}{\ln x}},\qquad x\geq 10{.}544{.}111}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de574cc895b4defe2b2723a3ef5f014286a67e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.439ex; height:4.843ex;" alt="{\displaystyle |\vartheta (x)-x|\leq 0{,}006788\,{\frac {x}{\ln x}},\qquad x\geq 10{.}544{.}111}" loading="lazy"></span></dd></dl>
<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 |\psi (x)-x|\leq 0{,}006409\,{\frac {x}{\ln x}},\qquad x\geq \exp(22)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mn>0,006</mn>
<mn>409</mn>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>x</mi>
<mo>≥<!-- ≥ --></mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>22</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\psi (x)-x|\leq 0{,}006409\,{\frac {x}{\ln x}},\qquad x\geq \exp(22)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f44f5a427da989b1aa4cc4230e62b5b7165820fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.672ex; height:4.843ex;" alt="{\displaystyle |\psi (x)-x|\leq 0{,}006409\,{\frac {x}{\ln x}},\qquad x\geq \exp(22)}" loading="lazy"></span></dd></dl>
<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 \psi (x)-\vartheta (x)<0{,}0000132\,{\frac {x}{\ln x}},\qquad x\geq \exp(30).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>ϑ<!-- ϑ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>&lt;</mo>
<mn>0,000</mn>
<mn>0132</mn>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>x</mi>
<mo>≥<!-- ≥ --></mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>30</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)-\vartheta (x)&lt;0{,}0000132\,{\frac {x}{\ln x}},\qquad x\geq \exp(30).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c493a4dc57fc393df2951a59731747ed8d9cca9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.37ex; height:4.843ex;" alt="{\displaystyle \psi (x)-\vartheta (x)<0{,}0000132\,{\frac {x}{\ln x}},\qquad x\geq \exp(30).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Verwandtschaft_der_beiden_Funktionen">Verwandtschaft der beiden Funktionen</h3></div>
<p>Es gilt
</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 \psi (x)=\sum _{p\leq x}k\log p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
</mrow>
</munder>
<mi>k</mi>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{p\leq x}k\log p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b13acdec2698b7719326d434d0f1411aacde5609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.619ex; height:5.843ex;" alt="{\displaystyle \psi (x)=\sum _{p\leq x}k\log p}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> <a href="Ganze_Zahl" title="Ganze Zahl">ganz</a> und dann durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}\leq x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{k}\leq x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ea179ae590bada353c833bfccfabbdf31dbd100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.776ex; height:3.009ex;" alt="{\displaystyle p^{k}\leq x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k+1}\geq x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>≥<!-- ≥ --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{k+1}\geq x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db02b33574545682e2631632faf0e15c4faecd6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.876ex; height:3.009ex;" alt="{\displaystyle p^{k+1}\geq x}" loading="lazy"></span> eindeutig bestimmt ist.
</p><p>Ein direkterer Zusammenhang entsteht durch
</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 \psi (x)=\sum _{n=1}^{\infty }\vartheta \left(x^{\frac {1}{n}}\right)=\sum _{n=1}^{\lfloor \log _{2}(x)\rfloor }\vartheta \left(x^{\frac {1}{n}}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</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>
</mrow>
</munderover>
<mi>ϑ<!-- ϑ --></mi>
<mrow>
<mo>(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
</msup>
<mo>)</mo>
</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">
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
</mrow>
</munderover>
<mi>ϑ<!-- ϑ --></mi>
<mrow>
<mo>(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta \left(x^{\frac {1}{n}}\right)=\sum _{n=1}^{\lfloor \log _{2}(x)\rfloor }\vartheta \left(x^{\frac {1}{n}}\right).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ac792dae53d2d70d261dde422947cab53f2d98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.095ex; height:7.676ex;" alt="{\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta \left(x^{\frac {1}{n}}\right)=\sum _{n=1}^{\lfloor \log _{2}(x)\rfloor }\vartheta \left(x^{\frac {1}{n}}\right).}" loading="lazy"></span></dd></dl>
<p>Man bemerke, dass <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 \vartheta \left(x^{\frac {1}{n}}\right)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϑ<!-- ϑ --></mi>
<mrow>
<mo>(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vartheta \left(x^{\frac {1}{n}}\right)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/379e5606bac3e0103e9d9e5e7a755540c0e2b762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.996ex; height:4.843ex;" alt="{\displaystyle \vartheta \left(x^{\frac {1}{n}}\right)=0}" loading="lazy"></span> 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 n\geq \log _{2}(x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<msub>
<mi>log</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq \log _{2}(x).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfab75725838ae4c0a7a1fc57198f9433d4ee3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.305ex; height:2.843ex;" alt="{\displaystyle n\geq \log _{2}(x).}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Die_„exakte_Formel“"><span id="Die_.E2.80.9Eexakte_Formel.E2.80.9C"></span>Die „exakte Formel“</h2></div>
<p>1895 bewies <a href="Hans_von_Mangoldt_(Mathematiker)" title="Hans von Mangoldt (Mathematiker)">Hans Karl Friedrich von Mangoldt</a> folgende Formel, die im Englischen auch als <i>„explicit formula“</i> bezeichnet wird:<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>
<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 \psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln(2\pi )-{\frac {1}{2}}\ln \left(1-x^{-2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ρ<!-- ρ --></mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ρ<!-- ρ --></mi>
</mrow>
</msup>
<mi>ρ<!-- ρ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln(2\pi )-{\frac {1}{2}}\ln \left(1-x^{-2}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b366c63cc40b43e09bb9abe55dd627785e31af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.955ex; height:6.676ex;" alt="{\displaystyle \psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln(2\pi )-{\frac {1}{2}}\ln \left(1-x^{-2}\right)}" loading="lazy"></span></dd></dl>
<p>Dabei ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&gt;</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x&gt;1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0549e1fb7ee2023519833093c6e3b60236e7d09f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>1}" loading="lazy"></span> und nicht prim oder eine Primzahlpotenz und die Summe läuft über alle nichttrivialen Nullstellen <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 \rho \,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho \,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1d651c28959a0f15127c097ff4488b123d9e708.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.589ex; height:2.176ex;" alt="{\displaystyle \rho \,}" loading="lazy"></span> der <a href="Riemannsche_Zeta-Funktion" title="Riemannsche Zeta-Funktion">Riemannschen Zeta-Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Referenzen">Referenzen</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">Pierre Dusart: <i>Sharper bounds for ψ, θ, π, p<sub>k</sub></i>. In: Rapport de recherche n° 1998-06, Université de Limoges. <a rel="nofollow" class="external text" href="https://www.unilim.fr/laco/rapports/1998/R1998_06.pdf">PDF</a></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/ExplicitFormula.html"><i>Explicit Formula</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch). </span>
</li>
</ol>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ChebyshevFunctions.html"><i>Chebyshev Function</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>
<li><a rel="nofollow" class="external text" href="https://planetmath.org/?op=getobj&amp;from=objects&amp;id=4020"><i>Mangoldt Summatory Function</i></a> und <a rel="nofollow" class="external text" href="https://planetmath.org/?op=getobj&amp;from=objects&amp;id=4573"><i>Chebyshev Function</i></a> auf <a href="PlanetMath" title="PlanetMath">PlanetMath</a></li>
<li><a href="Harold_Davenport" title="Harold Davenport">Harold Davenport</a>, <a href="Hugh_Montgomery_(Mathematiker)" title="Hugh Montgomery (Mathematiker)">Hugh L. Montgomery</a>: <i>Multiplicative number theory</i>. Springer Verlag 2000, ISBN 0-387-95097-4, ISBN 978-0-387-95097-6. <a rel="nofollow" class="external text" href="https://books.google.de/books?vid=ISBN0387950974&amp;id=U91lsCaJJmsC&amp;pg=PA104&amp;lpg=PA104&amp;sig=FhUIDFFTKNXSWhDM27PwfriD1gw&amp;redir_esc=y&amp;hl=de#v=onepage&amp;q=&amp;f=false">§. 17. GBS, eingeschränkt</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wikiversity"></span></span></div><b><a href="https://de.wikiversity.org/wiki/Kurs:Zahlentheorie_(Osnabr%C3%BCck_2008)/Vorlesung_12" class="extiw external" title="v:Kurs:Zahlentheorie (Osnabrück 2008)/Vorlesung 12">Wikiversity: Die Abschätzungen von Tschebyschow</a></b>&nbsp;– Kursmaterialien</div></div><!--htdig_noindex--><div><div class="zim-footer">
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