On the location of the zero-free half-plane of a random Epstein zeta function

TL;DR

This paper investigates the distribution of zeros of the Epstein zeta function for random high-dimensional lattices, establishing a limit distribution and providing explicit formulas for its characteristics.

Contribution

It introduces the first explicit limit distribution for the zeros of Epstein zeta functions of random lattices in high dimensions.

Findings

The supremum of the real parts of zeros converges to a limit distribution.

An explicit formula for the distribution function of this limit is derived.

The results deepen understanding of the zero distribution in high-dimensional lattice zeta functions.

Abstract

In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function E_n(L,s) and prove that this random variable has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function.