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abstract: 'In this paper we study the distribution of the non-trivial zeros of the zeta-function $\zeta(s)$ (and other L-functions) under Montgomery’s pair correlation approach. We use semidefinite programming to improve the asymptotic bounds for $N^*(T)$, $N_d(T)$ and $N({\lambda},T)$.'
address:
- 'IMPA - Instituto Nacional de Matemática Pura e Aplicada - Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil 22460-320'
- 'Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany'
- 'Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Simons building (Building 2), Room 2-241'
author:
- 'Andrés Chirre, Felipe Gonçalves and David de Laat'
title: 'Pair Correlation Estimates for the Zeros of the Zeta-Function via Semidefinite Programming'
---

Introduction
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In this paper we give improved asymptotic bounds for several quantities related to the zeros of the zeta-function (and other functions) under Montgomery’s pair correlation approach [@M]. The key idea is to replace the usual bandlimited auxiliary functions by the class of functions used in the linear programming bounds developed by Cohn and Elkies [@CE] for the sphere packing problem. The advantage of this framework is that it reduces the problems to convex optimization problems that can be solved numerically via semidefinite programming. 