Fourier Optimization, de Branges Spaces, and Zeros of L-functions

This thesis deals with a few topics at the intersection of Fourier analysis, number theory, and complex analysis. Using the framework of Fourier optimization we obtain new bounds related to the following questions in number theory: the least quadratic non-residue, the least prime in an arithmetic progression, and Montgomery's pair correlation conjecture. We also make contributions related to Hilbert spaces of entire functions, namely, studying norms of embeddings between weighted Paley--Wiener spaces, finding the sharp constant for an operator of multiplication in certain de Branges spaces, and introducing new sign uncertainty principles for functions of exponential type.