{
  "id": "1011.1825",
  "version": "v2",
  "published": "2010-11-08T15:21:51.000Z",
  "updated": "2011-01-18T12:38:26.000Z",
  "title": "Extreme values of the Dedekind $Ψ$ function",
  "authors": [
    "Patrick Solé",
    "Michel Planat"
  ],
  "comment": "5 pages, to appear in Journal of Combinatorics and Number theory",
  "journal": "Journal of Combinatorics and Number Theory 3, 1 (2011) 1-6",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Psi(n):=n\\prod_{p | n}(1+\\frac{1}{p})$ denote the Dedekind $\\Psi$ function. Define, for $n\\ge 3,$ the ratio $R(n):=\\frac{\\Psi(n)}{n\\log\\log n}.$ We prove unconditionally that $R(n)< e^\\gamma$ for $n\\ge 31.$ Let $N_n=2...p_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\\frac{e^\\gamma}{\\zeta(2)}$ for $n\\ge 3$ is equivalent to the Riemann Hypothesis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-18T12:38:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme values",
      "riemann hypothesis",
      "equivalent"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.1825S"
    }
  }
}