On the $q$-analogue of the pair correlation conjecture via Fourier optimization

TL;DR

This paper investigates the $q$-analogue of Montgomery's pair correlation function, providing bounds under GRH by extending Fourier analysis techniques and employing computational optimization methods.

Contribution

It introduces a novel Fourier analysis approach combined with non-smooth programming to bound the $q$-analogue of the pair correlation conjecture under GRH.

Findings

Bounds for the $q$-analogue average are close to the conjectured value of 1.

Extension of Fourier analysis methods to this problem.

Application of computational optimization techniques to number theory conjectures.

Abstract

We study the $q$-analogue of the average of Montgomery's function $F(\alpha, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of 1. To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.