Fourier optimization and pair correlation problems

Abstract

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery’s form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.