How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest – our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis! If we ask for ‘averaged’ results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this ‘square-root’ barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I’ll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.
This video is part of the Number Theory Web Seminar series.
