The twin prime conjecture asserts that there are infinitely many primes $p$ for which $p + 2$ is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer $h$ for which $p$ and $p + h$ are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the $q$-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.
Régis de la Bretèche
