Chebyshev theorems on prime numbers

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

Let be the number of primes not exceeding , let be an integer , let be a prime number, let be the natural logarithm of , and let


1) For any the sum of the series

has a finite limit as .

2) For arbitrary small and arbitrary large , the function satisfies the two inequalities

infinitely often.

3) The fraction cannot have a limit distinct from 1 as .

4) If can be expressed algebraically in , and up to order , then the expression must be (*). After this, Chebyshev introduced two new distribution functions for prime numbers — the Chebyshev functions (cf. Chebyshev function)

and actually determined the order of growth of these functions. Hence he was the first to obtain the order of growth of and of the -th prime number . More precisely, he proved:

5) If , then for the inequalities


6) For larger than some , the inequality


7) There exist constants such that for all the -th prime number satisfies the inequalities

8) For there is at least one prime number in the interval (Bertrand's postulate).

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

and their derivatives as . The basis of the method of deducing 5)–8) is the Chebyshev identity:


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


By now (1987) Chebyshev's theorems have been superceded by better results.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

(see [a1] for even better results); further changes sign infinitely often. More results, as well as references, can be found in [a1], Chapt. 12, Notes.


[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Chebyshev theorems on prime numbers. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article