Number Theory Seminar: Fourier optimization, prime gaps, and the least quadratic non-residue, by Micah Milinovich
April 18, 2025
1:00 PM - 1:50 PM
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 attempts to optimize a
certain quantity. I will describe how two such Fourier optimization
frameworks 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 in
analysis can be stated in accessible terms, but finding the exact
answer appears to be rather subtle. However, we can 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 (Lethbridge),
A. Ramos (SISSA), and K. Soundararajan (Stanford).
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Date posted
Apr 14, 2025
Date updated
Apr 14, 2025
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