Fourier optimization and pair correlation problems

TL;DR

This paper develops a Fourier analysis framework to bound pair correlation functions of various number-theoretic sequences, connecting these bounds to extremal problems and providing insights into conjectures related to the Riemann zeta function.

Contribution

It introduces a novel Fourier analysis approach to bound pair correlations for diverse sequences, linking these bounds to extremal problems and implications for conjectures.

Findings

Established bounds for pair correlation of zeros of L-functions

Connected pair correlation bounds to extremal Fourier problems

Provided evidence against certain conjectures in number theory

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.