Fourier optimization and consequences of the generalized Riemann hypothesis

TL;DR

This paper explores how Fourier optimization techniques relate to number theory problems, providing conditional bounds under the generalized Riemann hypothesis for prime gaps and quadratic non-residues.

Contribution

It introduces a Fourier optimization framework to derive new conditional bounds in number theory, connecting harmonic analysis with prime distribution problems.

Findings

Conditional bounds on prime gaps under GRH

Bounds on least quadratic non-residues under GRH

Framework linking Fourier analysis and number theory

Abstract

We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier optimization framework, on bounding the maximum possible gap between consecutive prime numbers represented by a given quadratic form; and on bounding the least quadratic non-residue modulo a prime number. This is based on joint works with Emanuel Carneiro, Andr\'es Chirre, Micah Milinovich, and Antonio Pedro Ramos.