Bombieri–Vinogradov theorem

In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) [E. Bombieri, "Le Grand Crible dans la Théorie Analytique des Nombres" (Seconde Édition). Astérisque 18, Paris 1987.] is a major result of analytic number theory, obtained in the mid-1960s. It is named for Enrico Bombieri and A. I. Vinogradov [A.I. Vinogradov. The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965), pages 903-934; Corrigendum. ibid. 30 (1966), pages 719-720. (Russian)] , who published on a related topic, the density hypothesis, in 1965.

This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of Yuri Linnik two decades earlier. Besides Bombieri, Klaus Roth was working in this area.

tatement of the Bombieri–Vinogradov theorem

Let "A" be any positive real number. Then:$sum\_\{qleq\; Q\}max\_\{1le\; ale\; qatop\; (a,q)=1\}left|psi(x;q,a)-\{xoverphi(q)\}\; ight|=Oleft(x^\{1/2\}Q(log\; x)^5\; ight),$if:$x^\{1/2\}log^\{-A\}xleq\; Qleq\; x^\{1/2\}.$

Here φ("q") is the Euler totient function, which is the number of summands for the modulus "q", and:$psi(x;q,a)=sum\_\{nle\; xatop\; nequiv\; amod\; q\}Lambda(n),$where $Lambda$ denotes the von Mangoldt function.

A verbal description of this result is that it addresses the error term in the prime number theorem for arithmetic progressions, averaged over the moduli "q" up to "Q". For a certain range of "Q", which are around √"x" if we neglect logarithmic factors, the error averaged is nearly as small as √"x". This is quite unobvious, and without the averaging is about of the strength of the Generalized Riemann Hypothesis (GRH).

ee also

*Vinogradov's theorem

References