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Analytic number theory
A branch of number theory. Analytic number theory deals with the problems of distribution of primes,
studies the behaviour of number-theoretic functions, and the theory of algebraic and transcendental
numbers.
Distribution of prime numbers.
a) The problem of the distribution of prime numbers (primes) is one of the most interesting and most
difficult problems in analytic number theory. The first result on the problem of the distribution of primes
was obtained by Euclid: There are infinitely many primes. Let be the number of primes not
exceeding . Euclid's theorem can then be formulated as follows: as . The
next step was due to . Chebyshev (1850), who proved that:
1) The quantity satisfies the inequalities
where ; .
2) If there exists a limit of as , this limit is one.
The problem of the existence of this limit was solved in 1896 by J. Hadamard and by . de la Vallée-
Poussin, who established thereby that
De la Vallée-Poussin demonstrated a much more general assertion, viz. if
then
where is an absolute constant (cf. de la Vallée-Poussin theorem). This problem was solved by
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methods of the theory of functions of plex variable. The problem of estimating is closely
connected with the problem on the behaviour of a certain function of plex variable, which was first
(1859) studied by B. Riemann and which is now known as Riemann's zeta-function. This function is
defined by
Even earlier (1737, 1749) the zeta-function with real -values was studied by L. Euler, who demonstrated
the identity specifying the connection between and prime numbers:
where the product is taken ov