{
  "id": "1109.1385",
  "version": "v1",
  "published": "2011-09-07T08:12:00.000Z",
  "updated": "2011-09-07T08:12:00.000Z",
  "title": "On the Rankin-Selberg problem in short intervals",
  "authors": [
    "Aleksandar Ivić"
  ],
  "comment": "12 pages",
  "journal": "Moscow Journal of Combinatorics and Number Theory, vol. {\\bf 2}, issue 3, 2012, 3-17",
  "categories": [
    "math.NT"
  ],
  "abstract": "If $$ \\Delta(x) \\;:=\\; \\sum_{n\\leqslant x}c_n - Cx\\qquad(C>0) $$ denotes the error term in the classical Rankin-Selberg problem, then we obtain a non-trivial upper bound for the mean square of $\\Delta(x+U) - \\Delta(x)$ for a certain range of $U = U(X)$. In particular, under the Lindel\\\"of hypothesis for $\\zeta(s)$, it is shown that $$ \\int_X^{2X} \\Bigl(\\Delta(x+U)-\\Delta(x)\\Bigr)^2\\,{\\roman d} x \\;\\ll_\\epsilon\\; X^{9/7+\\epsilon}U^{8/7}, $$ while under the Lindel\\\"of hypothesis for the Rankin-Selberg zeta-function the integral is bounded by $X^{1+\\epsilon}U^{4/3}$. An analogous result for the discrete second moment of $\\Delta(x+U)-\\Delta(x)$ also holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-07T08:12:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "44A15",
      "26A12"
    ],
    "keywords": [
      "short intervals",
      "discrete second moment",
      "non-trivial upper bound",
      "error term",
      "rankin-selberg zeta-function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.1385I"
    }
  }
}