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One of the two functions, of a positive argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218701.png" />, defined as follows:
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One of the two functions, of a positive argument $x$, defined as follows:
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218702.png" /></td> </tr></table>
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\theta(x) = \sum_{p \le x} \log p\,,\ \ \ \psi(x) = \sum_{p^m \le x} \log p \ .
 
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$$
The first sum is taken over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218703.png" />, and the second over all positive integer powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218704.png" /> of prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218705.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218706.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218707.png" /> can be expressed in terms of the [[Mangoldt function|Mangoldt function]]
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The first sum is taken over all prime numbers $p \le x$, and the second over all positive integer powers $m$ of prime numbers $p$ such that $p^m \le x$. The function $\psi(x)$ can be expressed in terms of the [[Mangoldt function|Mangoldt function]]
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218708.png" /></td> </tr></table>
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\psi(x) = \sum_{n \le x} \Lambda(n) \ .
 
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$$
It follows from the definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c0218709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187010.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187011.png" /> is equal to the product of all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187012.png" />, and that the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187013.png" /> is equal to the least common multiple of all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187014.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187016.png" /> are related by the identity
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It follows from the definitions of $\theta(x)$ and $\psi(x)$ that $e^{\theta(x)}$ is equal to the product of all prime numbers $p \le x$, and that the quantity $e^{\psi(x)}$ is equal to the least common multiple of all positive integers $n \le x$. The functions $\theta(x)$ and $\psi(x)$ are related by the identity
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187017.png" /></td> </tr></table>
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\psi(x) = \theta(x) + \theta(x^{1/2}) + \theta(x^{1/3}) + \cdots \ .
 +
$$
  
 
These functions are also closely connected with the function
 
These functions are also closely connected with the function
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$$
 +
\pi(x) = \sum_{p \le x} 1
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187018.png" /></td> </tr></table>
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which expresses the number of the prime numbers $p \le x$.
 
 
which expresses the number of the prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187019.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Mémoire sur les nombres premiers"  ''J. Math. Pures Appl.'' , '''17'''  (1852)  pp. 366–390  (Oeuvres, Vol. 1, pp. 51–70)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Mémoire sur les nombres premiers"  ''J. Math. Pures Appl.'' , '''17'''  (1852)  pp. 366–390  (Oeuvres, Vol. 1, pp. 51–70)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
For properties of the Chebyshev functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021870/c02187021.png" /> see [[#References|[a1]]], Chapt. 12.
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For properties of the Chebyshev functions $\theta(x)$ and $\psi(x)$ see [[#References|[a1]]], Chapt. 12.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 18:16, 18 October 2014

One of the two functions, of a positive argument $x$, defined as follows: $$ \theta(x) = \sum_{p \le x} \log p\,,\ \ \ \psi(x) = \sum_{p^m \le x} \log p \ . $$ The first sum is taken over all prime numbers $p \le x$, and the second over all positive integer powers $m$ of prime numbers $p$ such that $p^m \le x$. The function $\psi(x)$ can be expressed in terms of the Mangoldt function $$ \psi(x) = \sum_{n \le x} \Lambda(n) \ . $$ It follows from the definitions of $\theta(x)$ and $\psi(x)$ that $e^{\theta(x)}$ is equal to the product of all prime numbers $p \le x$, and that the quantity $e^{\psi(x)}$ is equal to the least common multiple of all positive integers $n \le x$. The functions $\theta(x)$ and $\psi(x)$ are related by the identity $$ \psi(x) = \theta(x) + \theta(x^{1/2}) + \theta(x^{1/3}) + \cdots \ . $$

These functions are also closely connected with the function $$ \pi(x) = \sum_{p \le x} 1 $$

which expresses the number of the prime numbers $p \le x$.

References

[1] P.L. Chebyshev, "Mémoire sur les nombres premiers" J. Math. Pures Appl. , 17 (1852) pp. 366–390 (Oeuvres, Vol. 1, pp. 51–70)


Comments

For properties of the Chebyshev functions $\theta(x)$ and $\psi(x)$ see [a1], Chapt. 12.

References

[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Chebyshev function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_function&oldid=12881
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article