{
  "id": "1311.4041",
  "version": "v2",
  "published": "2013-11-16T10:00:26.000Z",
  "updated": "2014-03-23T03:16:43.000Z",
  "title": "The Mean Square of Divisor Function",
  "authors": [
    "Chaohua Jia",
    "Ayyadurai Sankaranarayanan"
  ],
  "comment": "This is a revised version",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\\sum_{n\\leq x}d^2(n)=xP(\\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\\over 5}+\\epsilon}), $$ where $\\epsilon$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\\over 2}+\\epsilon}). $$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\\over 2}(\\log x)^5\\log\\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $$ E(x)=O(x^{1\\over 2}(\\log x)^5). $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-23T03:16:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisor function",
      "mean square",
      "direct application",
      "cubic polynomial",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4041J"
    }
  }
}