{
  "id": "math/0209070",
  "version": "v2",
  "published": "2002-09-06T15:56:37.000Z",
  "updated": "2002-10-04T00:06:14.000Z",
  "title": "Criteria for Irrationality of Euler's Constant",
  "authors": [
    "Jonathan Sondow"
  ],
  "comment": "12 pages, 1 figure, 2 tables, proofs shortened & typos fixed, revised version accepted by Proc. Amer. Math. Soc",
  "journal": "Proc. Amer. Math. Soc. 131 (2003) 3335-3344",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)} = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma is rational, then its denominator does not divide the product d(2n)Binomial(2n,n). We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact logS(n). A by-product is a rapidly converging asymptotic formula for gamma, used by P. Sebah to compute it correct to 18063 decimals.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-10-04T00:06:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "05A19"
    ],
    "keywords": [
      "eulers constant",
      "irrationality",
      "aperys theorem",
      "rational number",
      "fractional part"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......9070S"
    }
  }
}