JournalscmhVol. 95, No. 1pp. 183–209

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
On the universality of the Epstein zeta function cover
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We study universality properties of the Epstein zeta function En(L,s)E_n(L,s) for lattices LL of large dimension nn and suitable regions of complex numbers ss. Our main result is that, as nn\to\infty, En(L,s)E_n(L,s) is universal in the right half of the critical strip as LL varies over all nn-dimensional lattices LL. 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 nn\to\infty, En(L1,s)En(L2,s)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 (L1,L2)(L_1,L_2) varies over all pairs of nn-dimensional lattices. Finally, we prove a more classical universality result for En(L,s)E_n(L,s) in the ss-variable valid for almost all lattices LL of dimension nn. As part of the proof we obtain a strong bound of En(L,s)E_n(L,s) on the critical line that is subconvex for n5n\geq 5 and almost all nn-dimensional lattices LL.

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