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 be the number of primes not exceeding
, let
be an integer
, let
be a prime number, let
be the natural logarithm of
, and let
![]() | (*) |
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1) For any the sum of the series
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has a finite limit as .
2) For arbitrary small and arbitrary large
, the function
satisfies the two inequalities
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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)
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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
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hold.
6) For larger than some
, the inequality
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holds.
7) There exist constants such that for all
the
-th prime number
satisfies the inequalities
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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
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and their derivatives as . The basis of the method of deducing 5)–8) is the Chebyshev identity:
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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By now (1987) Chebyshev's theorems have been superceded by better results.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.,
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(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.
References
[a1] | A. Ivic, "The Riemann zeta-function" , Wiley (1985) |
Chebyshev theorems on prime numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_theorems_on_prime_numbers&oldid=16234