Page "Enrico Bombieri" Paragraph 1

The Bombieri – Vinogradov theorem is one of the major applications of the large sieve method.

It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case.

This result can sometimes substitute for the still-unproved generalized Riemann hypothesis.