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The theorems 1)–8) on the distribution of prime numbers, proved by P.L. Chebyshev [[#References|[1]]] in 1848–1850.
 
The theorems 1)–8) on the distribution of prime numbers, proved by P.L. Chebyshev [[#References|[1]]] in 1848–1850.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220101.png" /> be the number of primes not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220102.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220103.png" /> be an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220104.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220105.png" /> be a prime number, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220106.png" /> be the natural logarithm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c0220107.png" />, and let
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Let $\pi(x)$ be the number of primes not exceeding $x$, let $m$ be an integer $\geq0$, let $p$ be a prime number, let $\ln u$ be the natural logarithm of $u$, and let
  
<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/c022/c022010/c0220108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\operatorname{li}x=\int\limits_2^x\frac{dt}{\ln x}=\label{*}\tag{*}$$
  
<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/c022/c022010/c0220109.png" /></td> </tr></table>
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$$=\frac x{\ln x}+\dots+\frac{(n-1)!x}{\ln^nx}+O\left(\frac x{\ln^{n+1}x}\right).$$
  
1) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201010.png" /> the sum of the series
+
1) For any $m$ the sum of the series
  
<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/c022/c022010/c02201011.png" /></td> </tr></table>
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$$\sum_{n=2}^\infty\left(\pi(n)-\pi(n-1)-\frac1{\ln n}\right)\frac{\ln^mn}{n^s}$$
  
has a finite limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201012.png" />.
+
has a finite limit as $s\to1+$.
  
2) For arbitrary small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201013.png" /> and arbitrary large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201014.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201015.png" /> satisfies the two inequalities
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2) For arbitrary small $a>0$ and arbitrary large $m$, the function $\pi(x)$ satisfies the two inequalities
  
<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/c022/c022010/c02201016.png" /></td> </tr></table>
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$$\pi(x)>\operatorname{li}x-ax\ln^{-m}x,\quad\pi(x)<\operatorname{li}x+ax\ln^{-m}x$$
  
 
infinitely often.
 
infinitely often.
  
3) The fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201017.png" /> cannot have a limit distinct from 1 as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201018.png" />.
+
3) The fraction $(\pi(x)\ln x)/x$ cannot have a limit distinct from 1 as $x\to\infty$.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201019.png" /> can be expressed algebraically in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201022.png" /> up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201023.png" />, then the expression must be (*). After this, Chebyshev introduced two new distribution functions for prime numbers — the Chebyshev functions (cf. [[Chebyshev function|Chebyshev function]])
+
4) If $\pi(x)$ can be expressed algebraically in $x$, $\ln x$ and $e^x$ up to order $x\ln^{-n}x$, then the expression must be \eqref{*}. After this, Chebyshev introduced two new distribution functions for prime numbers — the Chebyshev functions (cf. [[Chebyshev function|Chebyshev function]])
  
<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/c022/c022010/c02201024.png" /></td> </tr></table>
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$$\theta(x)=\sum_{p\leq x}\ln p,\quad\psi(x)=\sum_{p^m\leq x}\ln p,$$
  
and actually determined the order of growth of these functions. Hence he was the first to obtain the order of growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201025.png" /> and of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201026.png" />-th prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201027.png" />. More precisely, he proved:
+
and actually determined the order of growth of these functions. Hence he was the first to obtain the order of growth of $\pi(x)$ and of the $n$-th prime number $P_n$. More precisely, he proved:
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201028.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201029.png" /> the inequalities
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5) If $A=\ln(2^{1/2}3^{1/3}5^{1/5}/30^{1/30})$, then for $x>1$ the inequalities
  
<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/c022/c022010/c02201030.png" /></td> </tr></table>
+
$$\psi(x)>Ax-\frac52\ln x-1,$$
  
<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/c022/c022010/c02201031.png" /></td> </tr></table>
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$$\psi(x)<\frac65Ax+\frac5{4\ln6}\ln^2x+\frac54\ln x+1,$$
  
 
hold.
 
hold.
  
6) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201032.png" /> larger than some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201033.png" />, the inequality
+
6) For $x$ larger than some $x_0$ the inequality
  
<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/c022/c022010/c02201034.png" /></td> </tr></table>
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$$0.9212\ldots<\frac{\pi(x)\ln x}{x}<1.1055\dots$$
  
 
holds.
 
holds.
  
7) There exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201035.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201036.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201037.png" />-th prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201038.png" /> satisfies the inequalities
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7) There exist constants $a>0,A>0$ such that for all $n=1,2,\dots,$ the $n$-th prime number $P_n$ satisfies the inequalities
  
<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/c022/c022010/c02201039.png" /></td> </tr></table>
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$$an\ln n<P_n<An\ln n.$$
  
8) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201040.png" /> there is at least one prime number in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201041.png" /> (Bertrand's postulate).
+
8) For $a>3$ there is at least one prime number in the interval $(a,2a-2)$ (Bertrand's postulate).
  
 
The main idea of the method of proof of 1)–4) lies in the study of the behaviour of the quantities
 
The main idea of the method of proof of 1)–4) lies in the study of the behaviour of the quantities
  
<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/c022/c022010/c02201042.png" /></td> </tr></table>
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$$\sum_{n=2}^\infty\frac1{n^{1+s}}-\frac1s,\quad\ln s-\sum\ln\left(1-\frac1{p^{1+s}}\right),$$
  
<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/c022/c022010/c02201043.png" /></td> </tr></table>
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$$\sum_p\ln\left(1-\frac1{p^{1+s}}\right)+\sum_p\frac1{p^{1+s}},$$
  
and their derivatives as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201044.png" />. The basis of the method of deducing 5)–8) is the Chebyshev identity:
+
and their derivatives as $s\to0+$. The basis of the method of deducing 5)–8) is the Chebyshev identity:
  
<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/c022/c022010/c02201045.png" /></td> </tr></table>
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$$\ln[x]!=\sum_{n\leq x}\psi\left(\frac xn\right).$$
  
 
====References====
 
====References====
Line 65: Line 67:
 
By now (1987) Chebyshev's theorems have been superceded by better results. E.g.,
 
By now (1987) Chebyshev's theorems have been superceded by better results. E.g.,
  
<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/c022/c022010/c02201046.png" /></td> </tr></table>
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$$\pi(x)=\operatorname{li}(x)+O(x\exp(-c\sqrt{\log x}))$$
  
(see [[#References|[a1]]] for even better results); further <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022010/c02201047.png" /> changes sign infinitely often. More results, as well as references, can be found in [[#References|[a1]]], Chapt. 12, Notes.
+
(see [[#References|[a1]]] for even better results); further $\pi(x)-\operatorname{li}(x)$ changes sign infinitely often. More results, as well as references, can be found in [[#References|[a1]]], Chapt. 12, Notes.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR></table>

Latest revision as of 16:58, 14 February 2020


The theorems 1)–8) on the distribution of prime numbers, proved by P.L. Chebyshev [1] in 1848–1850.

Let $\pi(x)$ be the number of primes not exceeding $x$, let $m$ be an integer $\geq0$, let $p$ be a prime number, let $\ln u$ be the natural logarithm of $u$, and let

$$\operatorname{li}x=\int\limits_2^x\frac{dt}{\ln x}=\label{*}\tag{*}$$

$$=\frac x{\ln x}+\dots+\frac{(n-1)!x}{\ln^nx}+O\left(\frac x{\ln^{n+1}x}\right).$$

1) For any $m$ the sum of the series

$$\sum_{n=2}^\infty\left(\pi(n)-\pi(n-1)-\frac1{\ln n}\right)\frac{\ln^mn}{n^s}$$

has a finite limit as $s\to1+$.

2) For arbitrary small $a>0$ and arbitrary large $m$, the function $\pi(x)$ satisfies the two inequalities

$$\pi(x)>\operatorname{li}x-ax\ln^{-m}x,\quad\pi(x)<\operatorname{li}x+ax\ln^{-m}x$$

infinitely often.

3) The fraction $(\pi(x)\ln x)/x$ cannot have a limit distinct from 1 as $x\to\infty$.

4) If $\pi(x)$ can be expressed algebraically in $x$, $\ln x$ and $e^x$ up to order $x\ln^{-n}x$, then the expression must be \eqref{*}. After this, Chebyshev introduced two new distribution functions for prime numbers — the Chebyshev functions (cf. Chebyshev function)

$$\theta(x)=\sum_{p\leq x}\ln p,\quad\psi(x)=\sum_{p^m\leq x}\ln p,$$

and actually determined the order of growth of these functions. Hence he was the first to obtain the order of growth of $\pi(x)$ and of the $n$-th prime number $P_n$. More precisely, he proved:

5) If $A=\ln(2^{1/2}3^{1/3}5^{1/5}/30^{1/30})$, then for $x>1$ the inequalities

$$\psi(x)>Ax-\frac52\ln x-1,$$

$$\psi(x)<\frac65Ax+\frac5{4\ln6}\ln^2x+\frac54\ln x+1,$$

hold.

6) For $x$ larger than some $x_0$ the inequality

$$0.9212\ldots<\frac{\pi(x)\ln x}{x}<1.1055\dots$$

holds.

7) There exist constants $a>0,A>0$ such that for all $n=1,2,\dots,$ the $n$-th prime number $P_n$ satisfies the inequalities

$$an\ln n<P_n<An\ln n.$$

8) For $a>3$ there is at least one prime number in the interval $(a,2a-2)$ (Bertrand's postulate).

The main idea of the method of proof of 1)–4) lies in the study of the behaviour of the quantities

$$\sum_{n=2}^\infty\frac1{n^{1+s}}-\frac1s,\quad\ln s-\sum\ln\left(1-\frac1{p^{1+s}}\right),$$

$$\sum_p\ln\left(1-\frac1{p^{1+s}}\right)+\sum_p\frac1{p^{1+s}},$$

and their derivatives as $s\to0+$. The basis of the method of deducing 5)–8) is the Chebyshev identity:

$$\ln[x]!=\sum_{n\leq x}\psi\left(\frac xn\right).$$

References

[1] P.L. Chebyshev, "Oeuvres de P.L. Tchebycheff" , 1–2 , Chelsea (1961) (Translated from Russian)


Comments

By now (1987) Chebyshev's theorems have been superceded by better results. E.g.,

$$\pi(x)=\operatorname{li}(x)+O(x\exp(-c\sqrt{\log x}))$$

(see [a1] for even better results); further $\pi(x)-\operatorname{li}(x)$ changes sign infinitely often. More results, as well as references, can be found in [a1], Chapt. 12, Notes.

References

[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Chebyshev theorems on prime numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_theorems_on_prime_numbers&oldid=16234
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article