On the universality of the Epstein zeta function

TL;DR

This paper demonstrates the universality of the Epstein zeta function for high-dimensional lattices, showing it can approximate a wide class of functions in certain regions as the dimension grows.

Contribution

It establishes the universality properties of the Epstein zeta function in high dimensions, including new results on the difference of two such functions and bounds on the critical line.

Findings

$E_n(L,s)$ is universal in the right half of the critical strip as $n o $

The difference $E_n(L_1,s)-E_n(L_2,s)$ is universal in the full half-plane for large $n$

A subconvex bound for $E_n(L,s)$ on the critical line for almost all lattices

Abstract

We study universality properties of the Epstein zeta function $E_n(L,s)$ for lattices $L$ of large dimension $n$ and suitable regions of complex numbers $s$. Our main result is that, as $n\to\infty$, $E_n(L,s)$ is universal in the right half of the critical strip as $L$ varies over all $n$-dimensional lattices $L$. The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as $n\to\infty$, $E_n(L_1,s)-E_n(L_2,s)$ is universal in the full half-plane to the right of the critical line as $(L_1,L_2)$ varies over all pairs of $n$-dimensional lattices. Finally, we prove a more classical universality result for $E_n(L,s)$ in the $s$-variable valid for almost all lattices $L$ of dimension $n$. As part of the proof we obtain a strong bound of $E_n(L,s)$ on the critical line that is subconvex for $n\geq 5$ and almost all $n$-dimensional lattices $L$.