We study universality properties of the Epstein zeta function for lattices of large dimension and suitable regions of complex numbers . Our main result is that, as , is universal in the right half of the critical strip as varies over all -dimensional lattices . 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 , is universal in the full half-plane to the right of the critical line as varies over all pairs of -dimensional lattices. Finally, we prove a more classical universality result for in the -variable valid for almost all lattices of dimension . As part of the proof we obtain a strong bound of on the critical line that is subconvex for and almost all -dimensional lattices .
Cite this article
Johan Andersson, Anders Södergren, On the universality of the Epstein zeta function. Comment. Math. Helv. 95 (2020), no. 1, pp. 183–209DOI 10.4171/CMH/485