Pair Correlation Estimates for the Zeros of the Zeta Function via Semidefinite Programming

TL;DR

This paper employs semidefinite programming to enhance bounds on the distribution of non-trivial zeros of the Riemann zeta-function, advancing understanding of zero spacing, multiplicities, and related properties.

Contribution

It introduces a novel application of semidefinite programming to improve bounds in the zero distribution of the zeta-function, surpassing previous asymptotic estimates.

Findings

Improved bounds on the proportion of distinct zeros

Enhanced estimates of small gaps between zeros

Refined sums involving zero multiplicities

Abstract

In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function $\zeta(s)$ (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of $\zeta(s)$, including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros.