{
  "id": "math/0411537",
  "version": "v1",
  "published": "2004-11-24T08:08:28.000Z",
  "updated": "2004-11-24T08:08:28.000Z",
  "title": "On the higher moments of the error term in the divisor problem",
  "authors": [
    "Aleksandar Ivić",
    "Patrick Sargos"
  ],
  "comment": "27 pages",
  "journal": "Illinois Journal of Mathematics 51(2007), 353-377.",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \\int_1^X \\Delta^3(x){\\rm d}x = BX^{7/4} + O_\\epsilon(X^{\\beta+\\epsilon}) \\qquad(B > 0) $$ and $$ \\int_1^X \\Delta^4(x){\\rm d}x = CX^2 + O_\\epsilon(X^{\\gamma+\\epsilon}) \\qquad(C > 0) $$ with $\\beta = 7/5, \\gamma = 23/12$. This improves on the values $\\beta = 47/28, \\gamma = 45/23$, due to K.-M. Tsang. A result on the integrals of $\\Delta^3(x)$ and $\\Delta^4(x)$ in short intervals is also proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-11-24T08:08:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11M06"
    ],
    "keywords": [
      "error term",
      "higher moments",
      "dirichlet divisor problem",
      "main results",
      "asymptotic formulas"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11537I"
    }
  }
}