Bombieri–Vinogradov theorem

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 first result of this kind was obtained by Mark Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri and A. I. Vinogradov, who published on a related topic, the density hypothesis, in 1965. Perhaps somewhat unjustly, Barban's name is often forgotten.

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.

Let ${\displaystyle x}$ and ${\displaystyle Q}$ be any two positive real numbers with

Then

Here ${\displaystyle \varphi (q)}$ is the Euler totient function, which is the number of summands for the modulus q, and

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