Primes in arithmetic progressions

Primes in arithmetic progressions

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James Maynard, University of Oxford

Register at: https://math.princeton.edu/minerva-2021

How many primes are there in an arithmetic progression $a\mod{q}$? Dirichlet's theorem shows that the primes are roughly equidistributed for $a$ coprime to $q$, and the Generalized Riemann Hypothesis (GRH) would imply this equidistribution occurs whenever $q$ is smaller than the square-root of the size of the primes. Unfortunately we don't know how to prove the GRH, but the Bombieri-Vinogradov Theorem shows that the GRH holds 'on average' and often serves as an adequate unconditional substitute.

We will talk about new work which extends the Bombieri-Vinogradov Theorem beyond the 'square-root barrier' implying equidistribution results for primes in regions out of direct reach of the Riemann Hypothesis, extending work of Bombieri,Friedlander,Fouvry,Iwaniec and Zhang. This rests on a fun combination of ideas from algebraic geometry, the spectral theory of automorphic forms, and classical analytic number theory.