{
  "id": "1508.05836",
  "version": "v1",
  "published": "2015-08-24T15:07:06.000Z",
  "updated": "2015-08-24T15:07:06.000Z",
  "title": "On the universality of the Epstein zeta function",
  "authors": [
    "Johan Andersson",
    "Anders Södergren"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "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 basic 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 $L$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-24T15:07:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "epstein zeta function",
      "dimensional lattices",
      "large dimension",
      "study universality properties",
      "strong uniform estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150805836A"
    }
  }
}