Pair correlation for Dedekind zeta functions of abelian extensions

TL;DR

This paper investigates the zeros of Dedekind zeta functions of abelian extensions, providing new bounds on zero simplicity and extending Montgomery's pair correlation approach to these functions.

Contribution

It introduces novel bounds on zero distributions of Dedekind zeta functions, utilizing semidefinite programming to optimize over Schwartz functions for the first time.

Findings

More than 45% of zeros are distinct for quadratic fields

Extended Montgomery's pair correlation approach to Dedekind zeta functions

Applied semidefinite programming to optimize bounds over Schwartz functions

Abstract

Here we study problems related to the proportions of zeros, especially simple and distinct zeros on the critical line, of Dedekind zeta functions. We obtain new bounds on a counting function that measures the discrepancy of the zeta functions from having all zeros simple. In particular, for quadratic number fields, we deduce that more than 45% of the zeros are distinct. This extends work based on Montgomery's pair correlation approach for the Riemann zeta function. Our optimization problems can be interpreted as interpolants between the pair correlation bound for the Riemann zeta function and the Cohn-Elkies sphere packing bound in dimension 1. We compute the bounds through optimization over Schwartz functions using semidefinite programming and also show how semidefinite programming can be used to optimize over functions with bounded support.