Bombieri–Vinogradov theorem

In mathematics, Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s. It is named for Enrico Bombieri and A. I. Vinogradov, 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.

Statement of the Bombieri–Vinogradov theorem

Let A be any positive real number. Then

$\sum _{q\leq Q}\max _{y\leq x}\max _{1\leq a\leq q \atop (a,q)=1}\left|\psi (x;q,a)-{x \over \phi (q)}\right|=O\left(x^{1/2}Q(\log x)^{5}\right),$

if

$x^{1/2}\log ^{-A}x\leq Q\leq x^{1/2}.$

Here φ(q) is the Euler totient function, which is the number of summands for the modulus q, and

$\psi (x;q,a)=\sum _{n\leq x \atop n\equiv a\mod q}\Lambda (n),$

where $\Lambda$ denotes the von Mangoldt function.

A verbal description of this result is that it addresses the error term in Dirichlet's theorem on 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).