Abstract: We will discuss several problems at the interface of analytic number theory and Fourier analysis, involving the distribution of values and zeros of the Riemann zeta-function, and also the distribution of integers and primes represented by quadratic forms. First, we discuss Selberg's celebrated central limit theorem, which roughly states that the logarithm of the Riemann zeta-function is normally distributed on the critical line. We will discuss related joint works with Andrés Chirre and with Meghann Moriah Lugar and Micah B. Milinovich. In particular, we discuss our proof of a conjecture of Berry (1988), conditional on RH and on a strong version of the pair correlation conjecture. Finally, we also discuss a Fourier analysis approach to the study of integers and primes represented by binary quadratic forms -- a classical problem going back to Fermat (joint work with Andrés Chirre).