{
  "id": "1711.09589",
  "version": "v1",
  "published": "2017-11-27T09:15:39.000Z",
  "updated": "2017-11-27T09:15:39.000Z",
  "title": "On certain integrals involving the Dirichlet divisor problem",
  "authors": [
    "Aleksandar Ivić",
    "Wenguang Zhai"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that $$ \\int_1^X\\Delta(x)\\Delta_3(x)\\,dx \\ll X^{13/9}\\log^{10/3}X, \\quad \\int_1^X\\Delta(x)\\Delta_4(x)\\,dx \\ll_\\varepsilon X^{25/16+\\varepsilon}, $$ where $\\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\\zeta^k(s)$ ($\\Delta_2(x) \\equiv \\Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\\Delta_k(x)$. We also obtain the analogues of the above bounds when $\\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\\zeta(1/2+it)|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-27T09:15:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N36",
      "11M06"
    ],
    "keywords": [
      "dirichlet divisor problem",
      "error term",
      "mean square formula",
      "mean square results",
      "asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}