{
  "id": "1105.2847",
  "version": "v1",
  "published": "2011-05-13T22:24:19.000Z",
  "updated": "2011-05-13T22:24:19.000Z",
  "title": "On the value distribution of the Epstein zeta function in the critical strip",
  "authors": [
    "Anders Södergren"
  ],
  "comment": "36 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the value distribution of the Epstein zeta function $E_n(L,s)$ for $0<s<\\frac{n}{2}$ and a random lattice $L$ of large dimension $n$. For any fixed $c\\in(1/4,1/2)$ and $n\\to\\infty$, we prove that the random variable $V_n^{-2c}E_n(\\cdot,cn)$ has a limit distribution, which we give explicitly (here $V_n$ is the volume of the $n$-dimensional unit ball). More generally, for any fixed $\\ve>0$ we determine the limit distribution of the random function $c\\mapsto V_n^{-2c}E_n(\\cdot,cn)$, $c\\in[1/4 +\\ve, 1/2-\\ve]$. After compensating for the pole at $c=\\frac12$ we even obtain a limit result on the whole interval $[\\frac14+\\ve,\\frac12]$, and as a special case we deduce the following strengthening of a result by Sarnak and Str\\\"ombergsson concerning the height function $h_n(L)$ of the flat torus $\\R^n/L$: The random variable $n\\big\\{h_n(L)-(\\log(4\\pi)-\\gamma+1)\\big\\}+\\log n$ has a limit distribution as $n\\to\\infty$, which we give explicitly. Finally we discuss a question posed by Sarnak and Str\\\"ombergsson as to whether there exists a lattice $L\\subset\\R^n$ for which $E_n(L,s)$ has no zeros in $(0,\\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-13T22:24:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E45"
    ],
    "keywords": [
      "epstein zeta function",
      "value distribution",
      "critical strip",
      "limit distribution",
      "dimensional unit ball"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.2847S"
    }
  }
}