{
  "id": "0903.5240",
  "version": "v3",
  "published": "2009-03-30T15:08:06.000Z",
  "updated": "2010-03-12T02:30:37.000Z",
  "title": "Transcendence of generating functions whose coefficients are multiplicative",
  "authors": [
    "Jason P. Bell",
    "Michael Coons"
  ],
  "comment": "This paper has been withdrawn and replaced with a more current version; see arXiv:1003.2221",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a field of characteristic 0, $f:\\mathbb{N}\\to K$ be a multiplicative function, and $F(z)=\\sum_{n\\geq 1} f(n)z^n\\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $\\chi(n)$ such that $f(n)=n^k \\chi(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, the generating function of a multiplicative function $f:\\mathbb{N}\\to K$ is either transcendental or rational.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-03-12T02:30:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N64",
      "11J91"
    ],
    "keywords": [
      "generating function",
      "transcendence",
      "coefficients",
      "natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.5240B"
    }
  }
}