Fourier optimization, the least quadratic non-residue, and the least prime in an arithmetic progression

By means of a Fourier optimization framework, we improve the current asymptotic bounds under the Generalized Riemann Hypothesis for two classical problems in number theory: the problem of estimating the least quadratic non-residue modulo a prime, and the problem of estimating the least prime in an arithmetic progression. In an appendix, we illustrate the connection between bounds for the argument of Dirichlet

L

-functions on the critical line and bounds for the least quadratic non-residue.