{
  "id": "1112.5072",
  "version": "v1",
  "published": "2011-12-21T16:06:24.000Z",
  "updated": "2011-12-21T16:06:24.000Z",
  "title": "On the rational approximation of the sum of the reciprocals of the Fermat numbers",
  "authors": [
    "Michael Coons"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\C{G}(z):=\\sum_{n=0}^\\infty z^{2^n}(1-z^{2^n})^{-1}$ denote the generating function of the ruler function, and $\\C{F}(z):=\\sum_{n=0}^\\infty z^{2^n}(1+z^{2^n})^{-1}$; note that the special value $\\C{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_n:=2^{2^n}+1$. The functions $\\C{F}(z)$ and $\\C{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\\C{F}(\\ga)$ and $\\C{G}(\\ga)$ are transcendental for all algebraic numbers $\\ga$ which satisfy $0<\\ga<1$. For a sequence $\\mathbf{u}$, denote the Hankel matrix $H_n^p(\\mathbf{u}):=(u({p+i+j-2}))_{1\\leqslant i,j\\leqslant n}$. Let $\\ga$ be a real number. The {\\em irrationality exponent} $\\mu(\\ga)$ is defined as the supremum of the set of real numbers $\\mu$ such that the inequality $|\\ga-p/q|<q^{-\\mu}$ has infinitely many solutions $(p,q)\\in\\B{Z}\\times\\B{N}.$ In this paper, we first prove that the determinants of $H_n^1(\\mathbf{g})$ and $H_n^1(\\mathbf{f})$ are nonzero for every $n\\geqslant 1$. We then use this result to prove that for $b\\geqslant 2$ the irrationality exponents $\\mu(\\C{F}(1/b))$ and $\\mu(\\C{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-21T16:06:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J82",
      "41A21"
    ],
    "keywords": [
      "fermat numbers",
      "rational approximation",
      "reciprocals",
      "irrationality exponent",
      "real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.5072C"
    }
  }
}