{
  "id": "0803.0132",
  "version": "v3",
  "published": "2008-03-02T17:06:00.000Z",
  "updated": "2009-01-15T08:26:51.000Z",
  "title": "On the mean square of the Riemann zeta-function in short intervals",
  "authors": [
    "Aleksandar Ivić"
  ],
  "comment": "19 pages",
  "journal": "Publications de l'Institut Math\\'ematique 85(99), 2009, 1-17",
  "categories": [
    "math.NT"
  ],
  "abstract": "It is proved that, for $T^\\epsilon\\le G = G(T) \\le {1\\over2}\\sqrt{T}$, $$ \\int_T^{2T}\\Bigl(I_1(t+G)-I_1(t)\\Bigr)^2 dt = TG\\sum_{j=0}^3a_j\\log^j \\Bigl({\\sqrt{T}\\over G}\\Bigr) + O_\\epsilon(T^{1+\\epsilon}+ T^{1/2+\\epsilon}G^2) $$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G) = {1\\over\\sqrt{\\pi}}\\int_{-\\infty}^\\infty |\\zeta(1/2+it+iu)|^{2k} {\\rm e}^{-(u/G)^2} du. $$ The generalizations to the mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-01-15T08:26:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11N37"
    ],
    "keywords": [
      "mean square",
      "riemann zeta-function",
      "short intervals",
      "fixed natural number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.0132I"
    }
  }
}