{
  "id": "0908.3253",
  "version": "v3",
  "published": "2009-08-22T14:04:34.000Z",
  "updated": "2014-02-05T02:22:42.000Z",
  "title": "On the possible exceptions for the transcendence of the log-gamma function at rational entries",
  "authors": [
    "F. M. S. Lima"
  ],
  "comment": "7 pages, 1 figure. Fully revised and shortened (02/05/2014)",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a recent work [JNT \\textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\\,\\log{\\Gamma(x)} + \\log{\\Gamma(1-x)}\\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \\emph{one} possible exception, but the proof presented there in that work is \\emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \\emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcendence of $\\,\\log{\\pi}$, a number whose irrationality is not proved yet. This has an interesting consequence for the transcendence of the product $\\,\\pi \\cdot e$, another number whose irrationality remains unproven.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-05T02:22:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J81",
      "11J86",
      "11J91"
    ],
    "keywords": [
      "rational entries",
      "log-gamma function",
      "transcendence",
      "irrationality remains unproven",
      "consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.3253L"
    }
  }
}