Primes in arithmetic progressions and semidefinite programming

Chirre, Andrés; Pereira Júnior, Valdir; de Laat, David

doi:10.1090/mcom/3638

Primes in arithmetic progressions and semidefinite programming
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by Andrés Chirre, Valdir José Pereira Júnior and David de Laat HTML | PDF

Math. Comp. 90 (2021), 2235-2246 Request permission

Abstract:

Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit formula over all Dirichlet characters modulo $q \geq 3$, and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.

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