Primes in arithmetic progressions to smooth moduli

For large x and coprime a and q, the arithmetic progression n = a mod q contains approximately pi(x)/phi(q) primes up to x. For which moduli q is this a good approximation? In this talk, I will focus on results for smooth moduli, which were a key ingredient in Zhang's proof of bounded gaps between primes and later work of Polymath. Following arguments of the Polymath project, I will sketch how better equidistribution estimates for primes in APs are linked to stronger bounds on the infimum limit of gaps between m consecutive primes. I will then show how a refinement of the q-van der Corput method can be used to improve on Polymath's equidistribution estimates and thus to obtain better bounds on short gaps between 3 or more primes.