Dalhousie-AARMS AAMP Seminar: Micah Milinovich (U. Mississippi)

Title: Fourier optimization, prime gaps, and zeta zeros

Abstract: There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe two ways these types of Fourier optimization problems can arise in the context of the explicit formula, which relates the primes to the zeros of the Riemann zeta-function. Using information from the zeros to study the primes, I will show how one can prove the strongest known estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. Using the explicit formula in the other direction, one can also use Fourier optimization to prove the strongest known conditional estimates for the number of zeta zeros in an interval on the critical line. This is based on joint works with E. Carneiro, V. Chandee, and K. Soundararajan.