Fourier optimization and consequences of the generalized Riemann hypothesis

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.