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Chebyshev function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ (x) or θ (x) is given by

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<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p}</annotation>

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</math><img src="./ba38682ec80afa73c29c41a52a63ea1976ee7305.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.882ex; height:5.843ex;" alt="{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p}" loading="lazy"></dd></dl>

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

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

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</math><img src="./79e4debd0ab1c6ce342d0172a7643733305c37bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }" loading="lazy"> denotes the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>, with the sum extending over all <a href="Prime_number" title="Prime number">prime numbers</a> p that are less than or equal to x.

The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all <a href="Prime_power" title="Prime power">prime powers</a> not exceeding x

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{k\in \mathbb {N} }\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,}</annotation>

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</math><img src="./c9616f859e7f64da9044ce816f81c71268c7df41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:52.316ex; height:6.343ex;" alt="{\displaystyle \psi (x)=\sum _{k\in \mathbb {N} }\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,}" loading="lazy"></dd></dl>

where Λ is the <a href="Von_Mangoldt_function" title="Von Mangoldt function">von Mangoldt function</a>. The Chebyshev functions, especially the second one ψ (x), are often used in <a href="Mathematical_proof" title="Mathematical proof">proofs</a> related to <a href="Prime_number" title="Prime number">prime numbers</a>, because it is typically simpler to work with them than with the <a href="Prime-counting_function" title="Prime-counting function">prime-counting function</a>, π (x) (see <a href="#The_exact_formula">the exact formula</a> below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the <a href="Prime_number_theorem" title="Prime number theorem">prime number theorem</a>.

Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:

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<annotation encoding="application/x-tex">{\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}f_{i}(x).}</annotation>

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</math><img src="./1461464a1079ac0f0c8fc396040752ca2c061e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.73ex; height:4.009ex;" alt="{\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}f_{i}(x).}" loading="lazy"><a href="#cite_note-JK-1">[1]</a></dd></dl>

By minimizing this function for different values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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

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</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy">, one obtains every point on a <a href="Pareto_front" title="Pareto front">Pareto front</a>, even in the nonconvex parts.<a href="#cite_note-JK-1">[1]</a> Often the functions to be minimized are not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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

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<annotation encoding="application/x-tex">{\displaystyle |f_{i}-z_{i}^{*}|}</annotation>

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</math><img src="./6f115f813332a3ad3160686cf061dfb6a5cd5e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.218ex; height:3.009ex;" alt="{\displaystyle |f_{i}-z_{i}^{*}|}" loading="lazy"> for some scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{i}^{*}}">

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

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</math><img src="./a892ddf5cd48d5274f39fe926216cd16718cdf2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.145ex; height:2.843ex;" alt="{\displaystyle z_{i}^{*}}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.}">

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<annotation encoding="application/x-tex">{\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.}</annotation>

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</math><img src="./4c00d5ca1dc5555b5d7a2da610f505afbc2e4217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.008ex; height:4.009ex;" alt="{\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.}" loading="lazy"><a href="#cite_note-2">[2]</a>

All three functions are named in honour of <a href="Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>.

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

The second Chebyshev function can be seen to be related to the first by writing it as

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{p\leq x}k\log p}</annotation>

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</math><img src="./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"></dd></dl>

where k is the unique <a href="Integer" title="Integer">integer</a> such that p k ≤ x and x < p k + 1. The values of k are given in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A206722" class="extiw external" title="oeis:A206722">A206722</a>. A more direct relationship is given by

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta {\big (}x^{\frac {1}{n}}{\big )}.}</annotation>

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</math><img src="./1566a1130f0faadeb9a32958a6bc74538429f931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.842ex; height:6.843ex;" alt="{\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta {\big (}x^{\frac {1}{n}}{\big )}.}" loading="lazy"></dd></dl>

This last sum has only a finite number of non-vanishing terms, as

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<annotation encoding="application/x-tex">{\displaystyle \vartheta {\big (}x^{\frac {1}{n}}{\big )}=0\quad {\text{for}}\quad n>\log _{2}x={\frac {\log x}{\log 2}}.}</annotation>

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</math><img src="./7e610a437b8baf72904d5b5f06925ac5588f6a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.9ex; height:5.843ex;" alt="{\displaystyle \vartheta {\big (}x^{\frac {1}{n}}{\big )}=0\quad {\text{for}}\quad n>\log _{2}x={\frac {\log x}{\log 2}}.}" loading="lazy"></dd></dl>

The second Chebyshev function is the logarithm of the <a href="Least_common_multiple" title="Least common multiple">least common multiple</a> of the integers from 1 to n.

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {lcm} (1,2,\dots ,n)=e^{\psi (n)}.}</annotation>

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</math><img src="./1916fa8ca96e5c9229db994d57202b2fce5ea94b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.753ex; height:3.343ex;" alt="{\displaystyle \operatorname {lcm} (1,2,\dots ,n)=e^{\psi (n)}.}" loading="lazy"></dd></dl>

Values of lcm(1, 2, ..., n) for the integer variable n are given at <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A003418" class="extiw external" title="oeis:A003418">A003418</a>.

<div class="mw-heading mw-heading2"><h2 id="Relationships_between_ψ(x)/x_and_ϑ(x)/x">Relationships between ψ(x)/x and ϑ(x)/x</h2></div>

The following <a href="Theorem" title="Theorem">theorem</a> relates the two quotients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\psi (x)}{x}}}">

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

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</math><img src="./4fd57ef5296cfe65aa1f01c24024cd48dcc88668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.488ex; height:5.676ex;" alt="{\displaystyle {\frac {\psi (x)}{x}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\vartheta (x)}{x}}}">

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

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</math><img src="./1a78d6d11d87efe13c22f93a67566f221929ed96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.349ex; height:5.676ex;" alt="{\displaystyle {\frac {\vartheta (x)}{x}}}" loading="lazy"> .<a href="#cite_note-3">[3]</a>

Theorem: For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

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

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<annotation encoding="application/x-tex">{\displaystyle 0\leq {\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\leq {\frac {(\log x)^{2}}{2{\sqrt {x}}\log 2}}.}</annotation>

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</math><img src="./e89ab3334fc973e5ecf9cf91da5b3b62e3ecebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.856ex; height:6.843ex;" alt="{\displaystyle 0\leq {\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\leq {\frac {(\log x)^{2}}{2{\sqrt {x}}\log 2}}.}" loading="lazy"></dd></dl>

This <a href="Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> implies that

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }\!\left({\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\right)\!=0.}</annotation>

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</math><img src="./c7a52e9c7571cd0902b92befdc60899cd02efcef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.846ex; height:6.343ex;" alt="{\displaystyle \lim _{x\to \infty }\!\left({\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\right)\!=0.}" loading="lazy"></dd></dl>

In other words, if one of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)/x}">

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

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</math><img src="./66e2077de5a3e58bf6fc2fd8ff7bf98cbbb92ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.144ex; height:2.843ex;" alt="{\displaystyle \psi (x)/x}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta (x)/x}">

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

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</math><img src="./6380960292063c8001f8a33c8bf8de61bc7c4ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.005ex; height:2.843ex;" alt="{\displaystyle \vartheta (x)/x}" loading="lazy"> tends to a <a href="Limit_of_a_function" title="Limit of a function">limit</a> then so does the other, and the two limits are equal.

Proof: Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\sum _{n\leq \log _{2}x}\vartheta (x^{1/n})}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{n\leq \log _{2}x}\vartheta (x^{1/n})}</annotation>

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</math><img src="./5304f2d58e80c8768c7bbf898f1b6f9148dc6aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.038ex; height:6.009ex;" alt="{\displaystyle \psi (x)=\sum _{n\leq \log _{2}x}\vartheta (x^{1/n})}" loading="lazy">, we find that

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<annotation encoding="application/x-tex">{\displaystyle 0\leq \psi (x)-\vartheta (x)=\sum _{2\leq n\leq \log _{2}x}\vartheta (x^{1/n}).}</annotation>

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</math><img src="./da067fe7b25c074f25611d87e98d9042d69acf64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:36.399ex; height:6.009ex;" alt="{\displaystyle 0\leq \psi (x)-\vartheta (x)=\sum _{2\leq n\leq \log _{2}x}\vartheta (x^{1/n}).}" loading="lazy"></dd></dl>

But from the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta (x)}">

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

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</math><img src="./b04e7560f5506682b65131b187839ab987efeeaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.513ex; height:2.843ex;" alt="{\displaystyle \vartheta (x)}" loading="lazy"> we have the trivial inequality

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<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)\leq \sum _{p\leq x}\log x\leq x\log x}</annotation>

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</math><img src="./179ef61df1f90986bcd02294b3280ef80aacdb4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.546ex; height:5.843ex;" alt="{\displaystyle \vartheta (x)\leq \sum _{p\leq x}\log x\leq x\log x}" loading="lazy"></dd></dl>

so

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0\leq \psi (x)-\vartheta (x)&\leq \sum _{2\leq n\leq \log _{2}x}x^{1/n}\log(x^{1/n})\\&\leq (\log _{2}x){\sqrt {x}}\log {\sqrt {x}}\\&={\frac {\log x}{\log 2}}{\frac {\sqrt {x}}{2}}\log x\\&={\frac {{\sqrt {x}}\,(\log x)^{2}}{2\log 2}}.\end{aligned}}}</annotation>

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</math><img src="./dfe62604496d0cc70997861f297d4bf82b854875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:42.681ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}0\leq \psi (x)-\vartheta (x)&\leq \sum _{2\leq n\leq \log _{2}x}x^{1/n}\log(x^{1/n})\\&\leq (\log _{2}x){\sqrt {x}}\log {\sqrt {x}}\\&={\frac {\log x}{\log 2}}{\frac {\sqrt {x}}{2}}\log x\\&={\frac {{\sqrt {x}}\,(\log x)^{2}}{2\log 2}}.\end{aligned}}}" loading="lazy"></dd></dl>

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

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</math><img src="./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"> to obtain the inequality in the theorem.

<div class="mw-heading mw-heading2"><h2 id="Asymptotics_and_bounds">Asymptotics and bounds</h2></div>

The following bounds are known for the Chebyshev functions:<a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Dusart1999">[1]</a><a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Dusart2010">[2]</a> (in these formulas pk is the kth prime number; p1 = 2, p2 = 3, etc.)

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\vartheta (p_{k})&\geq k\left(\log k+\log \log k-1+{\frac {\log \log k-2.050735}{\log k}}\right)&&{\text{for }}k\geq 10^{11},\\[8px]\vartheta (p_{k})&\leq k\left(\log k+\log \log k-1+{\frac {\log \log k-2}{\log k}}\right)&&{\text{for }}k\geq 198,\\[8px]|\vartheta (x)-x|&\leq 0.006788\,{\frac {x}{\log x}}&&{\text{for }}x\geq 10\,544\,111,\\[8px]|\psi (x)-x|&\leq 0.006409\,{\frac {x}{\log x}}&&{\text{for }}x\geq e^{22},\\[8px]0.9999{\sqrt {x}}&<\psi (x)-\vartheta (x)<1.00007{\sqrt {x}}+1.78{\sqrt[{3}]{x}}&&{\text{for }}x\geq 121.\end{aligned}}}</annotation>

</semantics>

</math><img src="./f7301be23d04a6fb8d72a32bdccccb1cc2f9596f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.136ex; margin-bottom: -0.202ex; width:84.718ex; height:33.843ex;" alt="{\displaystyle {\begin{aligned}\vartheta (p_{k})&\geq k\left(\log k+\log \log k-1+{\frac {\log \log k-2.050735}{\log k}}\right)&&{\text{for }}k\geq 10^{11},\\[8px]\vartheta (p_{k})&\leq k\left(\log k+\log \log k-1+{\frac {\log \log k-2}{\log k}}\right)&&{\text{for }}k\geq 198,\\[8px]|\vartheta (x)-x|&\leq 0.006788\,{\frac {x}{\log x}}&&{\text{for }}x\geq 10\,544\,111,\\[8px]|\psi (x)-x|&\leq 0.006409\,{\frac {x}{\log x}}&&{\text{for }}x\geq e^{22},\\[8px]0.9999{\sqrt {x}}&<\psi (x)-\vartheta (x)<1.00007{\sqrt {x}}+1.78{\sqrt[{3}]{x}}&&{\text{for }}x\geq 121.\end{aligned}}}" loading="lazy"></dd></dl>

Furthermore, under the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|\vartheta (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\\|\psi (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\end{aligned}}}</annotation>

</semantics>

</math><img src="./6daba7dfd9a169cb1287c469092ee70fd4184fe5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:23.625ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}|\vartheta (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\\|\psi (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\end{aligned}}}" loading="lazy"></dd></dl>

for any ε > 0.

Upper bounds exist for both ϑ (x) and ψ (x) such that<a href="#cite_note-4">[4]</a> <a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Dusart2010">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\vartheta (x)&<1.000028x\\\psi (x)&<1.03883x\end{aligned}}}</annotation>

</semantics>

</math><img src="./306b2d5d60af21a6f84fc0c4da5419e427fc1cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.616ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\vartheta (x)&<1.000028x\\\psi (x)&<1.03883x\end{aligned}}}" loading="lazy"></dd></dl>

for any x > 0.

An explanation of the constant 1.03883 is given at <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A206431" class="extiw external" title="oeis:A206431">A206431</a>.

<div class="mw-heading mw-heading2"><h2 id="The_exact_formula">The exact formula</h2></div>

In 1895, <a href="Hans_Carl_Friedrich_von_Mangoldt" title="Hans Carl Friedrich von Mangoldt">Hans Carl Friedrich von Mangoldt</a> proved<a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Dav104">[4]</a> an <a href="Explicit_formulae_(L-function)" class="mw-redirect" title="Explicit formulae (L-function)">explicit expression</a> for ψ (x) as a sum over the nontrivial <a href="Zero_of_a_function" title="Zero of a function">zeros</a> of the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\tfrac {1}{2}}\log(1-x^{-2}).}</annotation>

</semantics>

</math><img src="./898a00b652dbbafd14be45de7e152098fb3fa51b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.395ex; height:7.176ex;" alt="{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\tfrac {1}{2}}\log(1-x^{-2}).}" loading="lazy"></dd></dl>

(The numerical value of <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠ζ′ (0)/ζ (0)⁠ is log(2π).) Here ρ runs over the nontrivial zeros of the zeta function, and ψ0 is the same as ψ, except that at its <a href="Jump_discontinuity" class="mw-redirect" title="Jump discontinuity">jump discontinuities</a> (the prime powers) it takes the value halfway between the values to the left and the right:

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<mtext>otherwise.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)={\frac {1}{2}}\!\left(\sum _{n\leq x}\Lambda (n)+\sum _{n<x}\Lambda (n)\right)={\begin{cases}\psi (x)-{\tfrac {1}{2}}\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\dots \\\,\psi (x)&{\mbox{otherwise.}}\end{cases}}}</annotation>

</semantics>

</math><img src="./2cdbcf0b88a02f40b6e19b1bb9c74321b05cdf72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.864ex; height:7.509ex;" alt="{\displaystyle \psi _{0}(x)={\frac {1}{2}}\!\left(\sum _{n\leq x}\Lambda (n)+\sum _{n<x}\Lambda (n)\right)={\begin{cases}\psi (x)-{\tfrac {1}{2}}\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\dots \\\,\psi (x)&{\mbox{otherwise.}}\end{cases}}}" loading="lazy"></dd></dl>

From the <a href="Taylor_series" title="Taylor series">Taylor series</a> for the <a href="Natural_logarithm" title="Natural logarithm">logarithm</a>, the last term in the explicit formula can be understood as a summation of ⁠xω/ω⁠ over the trivial zeros of the zeta function, ω = −2, −4, −6, ..., i.e.

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}={\tfrac {1}{2}}\log \left(1-x^{-2}\right).}</annotation>

</semantics>

</math><img src="./4daea393ef9db26fbf81681f122469e7e4369af0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.655ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}={\tfrac {1}{2}}\log \left(1-x^{-2}\right).}" loading="lazy"></dd></dl>

Similarly, the first term, x = ⁠x1/1⁠, corresponds to the simple <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> of the zeta function at 1. It being a pole rather than a zero accounts for the opposite sign of the term.

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

A theorem due to <a href="Erhard_Schmidt" title="Erhard Schmidt">Erhard Schmidt</a> states that, for some explicit positive constant K, there are infinitely many <a href="Natural_number" title="Natural number">natural numbers</a> x such that

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

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</math><img src="./81dc7cca021574896fabdd77f611713163d783f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.06ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x<-K{\sqrt {x}}}" loading="lazy"></dd></dl>

and infinitely many natural numbers x such that

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

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</math><img src="./4d5c56ce698b57dca44a39808a73be5a8faf5c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.899ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x>K{\sqrt {x}}.}" loading="lazy"><a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Sch03">[5]</a><a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Hard16">[6]</a></dd></dl>

In <a href="Big-O_notation" class="mw-redirect" title="Big-O notation">little-o notation</a>, one may write the above as

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\right).}</annotation>

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</math><img src="./c1382edbe8684f3610a75f16c766ec1b592b96e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.931ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\right).}" loading="lazy"></dd></dl>

<a href="G._H._Hardy" title="G. H. Hardy">Hardy</a> and <a href="J._E._Littlewood" class="mw-redirect" title="J. E. Littlewood">Littlewood</a><a class="external autonumber external" href="https://en.wikipedia.org/wiki/Chebyshev_function#endnote_Hard16">[7]</a> prove the stronger result, that

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\log \log \log x\right).}</annotation>

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</math><img src="./b579ef187d4523d0999c76e2c0974e1fc25610a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.725ex; height:3.009ex;" alt="{\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\log \log \log x\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_primorials">Relation to primorials</h2></div>

The first Chebyshev function is the logarithm of the <a href="Primorial" title="Primorial">primorial</a> of x, denoted x #:

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<annotation encoding="application/x-tex">{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p=\log \prod _{p\leq x}p=\log \left(x\#\right).}</annotation>

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</math><img src="./e15ceb51adf31c0cee64179d5e5ace53158233ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.733ex; height:5.843ex;" alt="{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p=\log \prod _{p\leq x}p=\log \left(x\#\right).}" loading="lazy"></dd></dl>

This proves that the primorial x # is asymptotically equal to e(1 + o(1))x, where "o" is the little-o notation (see <a href="Big_O_notation" title="Big O notation">big O notation</a>) and together with the prime number theorem establishes the asymptotic behavior of pn #.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_prime-counting_function">Relation to the prime-counting function</h2></div>

The Chebyshev function can be related to the prime-counting function as follows. Define

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<annotation encoding="application/x-tex">{\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}</annotation>

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</math><img src="./87f09f36af683ce08ee290af00b923f4c2a74020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.023ex; height:7.009ex;" alt="{\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}" loading="lazy"></dd></dl>

Then

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<annotation encoding="application/x-tex">{\displaystyle \Pi (x)=\sum _{n\leq x}\Lambda (n)\int _{n}^{x}{\frac {dt}{t\log ^{2}t}}+{\frac {1}{\log x}}\sum _{n\leq x}\Lambda (n)=\int _{2}^{x}{\frac {\psi (t)\,dt}{t\log ^{2}t}}+{\frac {\psi (x)}{\log x}}.}</annotation>

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</math><img src="./b6427253d033f913032ce0ba2afec714d14a2dd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:69.152ex; height:7.009ex;" alt="{\displaystyle \Pi (x)=\sum _{n\leq x}\Lambda (n)\int _{n}^{x}{\frac {dt}{t\log ^{2}t}}+{\frac {1}{\log x}}\sum _{n\leq x}\Lambda (n)=\int _{2}^{x}{\frac {\psi (t)\,dt}{t\log ^{2}t}}+{\frac {\psi (x)}{\log x}}.}" loading="lazy"></dd></dl>

The transition from Π to the <a href="Prime-counting_function" title="Prime-counting function">prime-counting function</a>, π, is made through the equation

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<annotation encoding="application/x-tex">{\displaystyle \Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi \left({\sqrt {x}}\,\right)+{\tfrac {1}{3}}\pi \left({\sqrt[{3}]{x}}\,\right)+\cdots }</annotation>

</semantics>

</math><img src="./366ea3365a925e84a1dd8555ba2d8d27aac1207e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.374ex; height:3.676ex;" alt="{\displaystyle \Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi \left({\sqrt {x}}\,\right)+{\tfrac {1}{3}}\pi \left({\sqrt[{3}]{x}}\,\right)+\cdots }" loading="lazy"></dd></dl>

Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form

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<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\Pi (x)+O\left({\sqrt {x}}\,\right).}</annotation>

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</math><img src="./98e847829ac41860a74018de325ce5c368249ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.948ex; height:3.009ex;" alt="{\displaystyle \pi (x)=\Pi (x)+O\left({\sqrt {x}}\,\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_Riemann_hypothesis">The Riemann hypothesis</h2></div>

The <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> states that all nontrivial <a href="Zero_of_a_function" title="Zero of a function">zeros</a> of the zeta function have <a href="Real_part" class="mw-redirect" title="Real part">real part</a> ⁠1/2⁠. In this case, |x ρ| = √x, and it can be shown that

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<annotation encoding="application/x-tex">{\displaystyle \sum _{\rho }{\frac {x^{\rho }}{\rho }}=O\!\left({\sqrt {x}}\,\log ^{2}x\right).}</annotation>

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</math><img src="./a7d45059a99d4d078acdb67cde090c9bf938be6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.808ex; height:6.676ex;" alt="{\displaystyle \sum _{\rho }{\frac {x^{\rho }}{\rho }}=O\!\left({\sqrt {x}}\,\log ^{2}x\right).}" loading="lazy"></dd></dl>

By the above, this implies

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<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\operatorname {li} (x)+O\!\left({\sqrt {x}}\,\log x\right).}</annotation>

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</math><img src="./c3fe97cec4c334d7f6ad8a45ea39133ad0f9c5a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.187ex; height:3.009ex;" alt="{\displaystyle \pi (x)=\operatorname {li} (x)+O\!\left({\sqrt {x}}\,\log x\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Smoothing_function">Smoothing function</h2></div>

The smoothing function is defined as

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}(x)=\int _{0}^{x}\psi (t)\,dt.}</annotation>

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</math><img src="./40d9a64670c4d85490e2555d95093a201534b158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.08ex; height:5.843ex;" alt="{\displaystyle \psi _{1}(x)=\int _{0}^{x}\psi (t)\,dt.}" loading="lazy"></dd></dl>

Obviously <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}(x)\sim {\frac {x^{2}}{2}}.}">

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</msub>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1}(x)\sim {\frac {x^{2}}{2}}.}</annotation>

</semantics>

</math><img src="./908203191fead1c3171043ca7991edc32019a7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.672ex; height:5.676ex;" alt="{\displaystyle \psi _{1}(x)\sim {\frac {x^{2}}{2}}.}" loading="lazy">

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

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFJoshua_Knowles2014" class="citation web cs1">Joshua Knowles (2 May 2014). <a rel="nofollow" class="external text" href="http://syllabus.cs.manchester.ac.uk/pgt/2017/COMP60342/COMP60342-2014-MOO.pdf">"Multiobjective Optimization Concepts, Algorithms and Performance Measures"</a> (PDF). The University of Manchester. p. 34.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHo-HuuHartjesVisserCurran2018" class="citation journal cs1">Ho-Huu, V.; Hartjes, S.; Visser, H. G.; Curran, R. (2018). <a rel="nofollow" class="external text" href="https://pure.tudelft.nl/ws/portalfiles/portal/30882193/FinalRevised_Improved_MOEA_D_for_BOPs_with_complicated_PFs.pdf">"An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization"</a> (PDF). Expert Systems with Applications. Delft University of Technology. Page 6 equation (2). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.eswa.2017.09.051">10.1016/j.eswa.2017.09.051</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFApostol2010" class="citation book cs1">Apostol, Tom M. (2010). Introduction to Analytic Number Theory. Springer. pp. 75–76.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFRosserSchoenfeld1962" class="citation journal cs1"><a href="J._Barkley_Rosser" title="J. Barkley Rosser">Rosser, J. Barkley</a>; <a href="Lowell_Schoenfeld" title="Lowell Schoenfeld">Schoenfeld, Lowell</a> (1962). <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255631807">"Approximate formulas for some functions of prime numbers"</a>. Illinois J. Math. 6: 64–94.</cite>

</li>

</ol></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}

/* end https://en.wikipedia.org/ */

</style><a href="#ref_Dusart2010">^</a> <a href="Pierre_Dusart" title="Pierre Dusart">Pierre Dusart</a>, "Estimates of some functions over primes without R.H.". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.0442">1002.0442</a></li>

<li><a href="#ref_Dusart1999">^</a> Pierre Dusart, "Sharper bounds for ψ, θ, π, pk", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(log k + log log k − 1) for k ≥ 2", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.</li>

<li><a href="#ref_Sch03">^</a> Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.</li>

<li><a href="#ref_Hard16">^</a> G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41 (1916) pp. 119–196.</li>

<li><a href="#ref_Dav104">^</a> <a href="Harold_Davenport" title="Harold Davenport">Davenport, Harold</a> (2000). In <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U91lsCaJJmsC&pg=PA104">Multiplicative Number Theory</a>. Springer. p. 104. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95097-4</bdi>. Google Book Search.</li></ul>

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

<ul><li><cite id="CITEREFApostol1976" class="citation cs2"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ChebyshevFunctions.html">"Chebyshev functions"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/MangoldtSummatoryFunction">"Mangoldt summatory function"</a>. <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</cite></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/ChebyshevFunctions">"Chebyshev functions"</a>. <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.math.ucsb.edu/~stopple/explicit.html">Riemann's Explicit Formula</a>, with images and movies</li></ul></div><div><div class="zim-footer">

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