{
  "id": "0707.1756",
  "version": "v5",
  "published": "2007-07-12T09:33:59.000Z",
  "updated": "2007-11-08T08:55:24.000Z",
  "title": "On the divisor function and the Riemann zeta-function in short intervals",
  "authors": [
    "Aleksandar Ivic"
  ],
  "comment": "18 pages",
  "journal": "Ramanujan J. 19(2009), 207-224",
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain, for $T^\\epsilon \\le U=U(T)\\le T^{1/2-\\epsilon}$, asymptotic formulas for $$ \\int_T^{2T}(E(t+U) - E(t))^2 dt,\\quad \\int_T^{2T}(\\Delta(t+U) - \\Delta(t))^2 dt, $$ where $\\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\\zeta(1/2+it)|$. Upper bounds of the form $O_\\epsilon(T^{1+\\epsilon}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \\le U =U(T) \\ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|\\zeta(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2007-11-08T08:55:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11N37"
    ],
    "keywords": [
      "riemann zeta-function",
      "short intervals",
      "divisor function",
      "number-theoretic error terms",
      "mean square formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1756I"
    }
  }
}