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

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.