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 ≥ 3, and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.

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In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of the zeta-function, including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros.@en