Julia Stadlmann: Primes in arithmetic progressions to smooth moduli

Abstract:

For coprime \(a\) and \(q\), the arithmetic progression \(n = a \bmod q\) contains approximately \(\pi(x)/\varphi(q)\) primes up to \(x\). When is this a good estimate? In many works in sieve theory, bounds on the error terms for primes in APs and averages of errors over certain moduli play an important role. In this talk, I will focus on smooth moduli, which were a key ingredient in Zhang’s proof of bounded gaps between primes.

Following arguments of the Polymath project, I will sketch how better equidistribution estimates are linked to stronger bounds on the infimum limit of gaps between m consecutive primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of Polymath for primes in APs to smooth moduli.