Fourier optimization and Montgomery's pair correlation conjecture

TL;DR

This paper employs Fourier optimization techniques to refine bounds on Montgomery's function related to the pair correlation of Riemann zeta zeros, assuming the Riemann hypothesis, and extends these results to Dirichlet L-functions.

Contribution

It introduces new averaging mechanisms and utilizes a broader class of test functions for improved bounds in pair correlation problems.

Findings

Bounds for F(α, T) are tightened to between 0.9303 and 1.3208.

Method extends to bounds for zeros of Dirichlet L-functions.

Assumes the Riemann hypothesis for results.

Abstract

Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery's function $F(\alpha, T)$ over long intervals by means of a Fourier optimization framework. The function $F(\alpha, T)$ is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach: (i) the introduction of new averaging mechanisms in our conceptual framework and (ii) the full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions. We conclude that such an average value, that is conjectured to be $1$, lies between $0.9303$ and $1.3208$. Our Fourier optimization framework also yields an improvement on the current bounds for the analogous problem concerning the non-trivial zeros in the family of Dirichlet $L$-functions.