{
  "id": "1110.5078",
  "version": "v2",
  "published": "2011-10-23T19:34:19.000Z",
  "updated": "2012-01-12T22:49:03.000Z",
  "title": "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis",
  "authors": [
    "Geoffrey Caveney",
    "Jean-Louis Nicolas",
    "Jonathan Sondow"
  ],
  "comment": "11 pages, 1 table, clarified Proposition 4, added reference 4",
  "journal": "Integers 11 (2011) article A33",
  "categories": [
    "math.NT",
    "math.HO"
  ],
  "abstract": "For n>1, let G(n)=\\sigma(n)/(n log log n), where \\sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \\ge \\max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erd\\H{o}s's results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-12T22:49:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11A41",
      "11Y55"
    ],
    "keywords": [
      "riemann hypothesis",
      "robins theorem",
      "elementary reformulation",
      "log log",
      "superabundant numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.5078C"
    }
  }
}