Chebyshev theorems on prime numbers

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)