Bombieri–Vinogradov theorem

The Bombieri–Vinogradov theorem bounds the average error in the prime number theorem for arithmetic progressions across moduli up to nearly the square root of the range. It gives, on average, distribution as strong as the generalized Riemann hypothesis predicts for individual moduli.

Prime number theorem for arithmetic progressions: A theorem describing how primes are distributed among residue classes coprime to a fixed modulus. The theorem averages the error in this result over many moduli.
Arithmetic progression: A sequence of integers formed by repeatedly adding a fixed common difference. Primes are counted separately in residue classes that form arithmetic progressions.
Enrico Bombieri: An Italian mathematician known for major contributions to analytic number theory and the large sieve. His work established one side of the theorem independently.
Brun–Titchmarsh theorem: An upper bound for the number of primes in an arithmetic progression, subject to a range condition on its modulus. It complements average distribution with useful bounds for individual progression counts.
Elliott–Halberstam conjecture: A conjecture asserting a higher level of distribution for primes than the Bombieri–Vinogradov theorem provides. It predicts average control beyond the square-root barrier.
Siegel–Walfisz theorem: A uniform prime-distribution theorem for arithmetic progressions whose moduli grow no faster than powers of the logarithm. It supplies the small-modulus range where uniform estimates are already available.
Euler totient function: The function counting residue classes modulo an integer that are relatively prime to it. It gives the expected share of primes in each admissible residue class.
A. I. Vinogradov: A Soviet mathematician whose work included foundational estimates for exponential sums and prime distribution. His independent proof gave the theorem its second namesake.
Maynard–Tao sieve: A sieve method that proves bounded gaps between primes by detecting several primes in admissible tuples. Bombieri–Vinogradov supplies distribution estimates used in bounded-gap arguments.
Generalized Elliott–Halberstam conjecture: A strengthening of the Elliott–Halberstam conjecture covering more general sequences and correlations. It extends the conjectured distribution framework beyond primes alone.

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