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Thursday, November 29, 2018

**Abstract:** I will discuss the proofs of the strongest known asymptotic and explicit estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. These proofs involve three main ingredients: the explicit formula relating the primes to the zeros of the Riemann zeta-function, the size of the constant in the Brun-Titchmarsh inequality, and Fourier optimization. I will also discuss how to use Fourier optimization, in particular the solution to the Beurling-Selberg extremal problem for the Poisson kernel, to estimate the variance of primes on short intervals sharpening the previous results of Selberg, Montgomery, Gallagher and Mueller, Goldston and Gonek, and others. This talk is based on joint works with E. Carneiro, V. Chandee, A. Chirre, and K. Soundararajan.