{
  "id": "math/0408022",
  "version": "v1",
  "published": "2004-08-02T14:58:31.000Z",
  "updated": "2004-08-02T14:58:31.000Z",
  "title": "The moments of the Riemann zeta-function. Part I: The fourth moment off the critical line",
  "authors": [
    "Aleksandar Ivić",
    "Yoichi Motohashi"
  ],
  "comment": "50 pages",
  "journal": "Functiones et Approximatio 35(2006), 133-181.",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \\int_{-\\infty}^\\infty|\\zeta(\\s+it)|^4g(t){\\rm d}t\\qquad(\\hf < \\sigma < 1 {\\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function $$\\eqalign{E_2(T,\\sigma) &: =\\int_0^T|\\zeta(\\sigma+it)|^4{rm d}t - {\\zeta^4(2\\sigma)\\over\\zeta(4\\sigma)}T -{T\\over3-4\\sigma}{({T\\over2\\pi} )}^{2-4\\sigma}{\\zeta^4(2-2\\sigma)\\over\\zeta(4-4\\sigma)}\\cr& - T^{2-2\\sigma}(a_0(\\sigma) + a_1(\\sigma)\\log T + a_2(\\sigma)\\log^2T),\\cr} $$ the error term in the asymptotic formula for the fourth moment of $|\\zeta(\\sigma+it)|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-08-02T14:58:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11F72",
      "11F66"
    ],
    "keywords": [
      "fourth moment",
      "riemann zeta-function",
      "critical line",
      "first part",
      "asymptotic formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8022I"
    }
  }
}