{
  "id": "math/0507231",
  "version": "v1",
  "published": "2005-07-12T08:45:53.000Z",
  "updated": "2005-07-12T08:45:53.000Z",
  "title": "A family of criteria for irrationality of Euler's constant",
  "authors": [
    "Marc Prévost"
  ],
  "journal": "Pr\\~A\\c{opyright}vost, Marc Legendre modified moments for Euler's constant. J. Comput. Appl. Math. 219 (2008), no. 2, 484--492",
  "categories": [
    "math.NT"
  ],
  "abstract": "Following earlier results of Sondow, we propose another criterion of irrationality for Euler's constant $\\gamma$. It involves similar linear combinations of logarithm numbers $L\\_{n,m}$. To prove that $\\gamma$ is irrational, it suffices to prove that, for some fixed $m$, the distance of $d\\_n L\\_{n,m}$ ($d\\_n$ is the least common multiple of the $n$ first integers) to the set of integers $\\mathbf{Z}$ does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior of $d\\_n$ is an obstacle to verify this sufficient condition. All the proofs use in a large manner the theory of Pad\\'e approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-07-12T08:45:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "41A21"
    ],
    "keywords": [
      "eulers constant",
      "irrationality",
      "sufficient condition",
      "rational numbers",
      "similar linear combinations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7231P"
    }
  }
}