In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli.
The Bombieri–Vinogradov theorem is named after Enrico Bombieri[2] and A. I. Vinogradov,[3] 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.
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.
This is not obvious, and without the averaging is about of the strength of the Generalized Riemann Hypothesis (GRH).