{
  "id": "0806.1694",
  "version": "v1",
  "published": "2008-06-10T15:38:18.000Z",
  "updated": "2008-06-10T15:38:18.000Z",
  "title": "Transcendence of the Gaussian Liouville number and relatives",
  "authors": [
    "Peter Borwein",
    "Michael Coons"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "{\\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\\gl(n))$; here $\\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\\em the Gaussian Liouville number} by $$\\gamma:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number $$\\sum_{k=0}^\\infty\\frac{2^{3^k}}{2^{3^k2}+2^{3^k}+1}=0.101100101101100100101...,$$ where the $n$th bit is determined by the parity of the number of prime divisors that are equivalent to 2 modulo 3. We use methods similar to that of Dekking's proof of the transcendence of the Thue--Morse number \\cite{Dek1} as well as a theorem of Mahler's \\cite{Mahl1}. (For completeness we provide proofs of all needed results.) This method involves proving the transcendence of formal power series arising as generating functions of completely multiplicative functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-10T15:38:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J81",
      "11A05"
    ],
    "keywords": [
      "gaussian liouville number",
      "transcendence",
      "prime divisors",
      "formal power series",
      "th bit reflects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1694B"
    }
  }
}