Mathematics Colloquia and Seminars

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For large $x$ and coprime $a$ and $q$, the arithmetic progression $n \equiv a \bmod q$ 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.