Micah Milinovich, University of Mississippi
"Fourier optimization, prime gaps, and the least quadratic non-residue"
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 such Fourier optimization frameworks that can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems can be stated in accessible terms, but finding the exact answer appears to be rather subtle. Instead, we experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (TU Graz), A. Ramos (SISSA), and K. Soundararajan (Stanford).
MC 5417