# On the universality of the Epstein zeta function

### Johan Andersson

Örebro University, Sweden### Anders Södergren

Chalmers University of Technology and Göteburg University, Göteborg, Sweden and University of Copenhagen, Denmark

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## 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 a novel combination of an approximation result for Dirichlet polynomials, 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 $L$.

## Cite this article

Johan Andersson, Anders Södergren, On the universality of the Epstein zeta function. Comment. Math. Helv. 95 (2020), no. 1, pp. 183–209

DOI 10.4171/CMH/485