{
  "id": "1003.2221",
  "version": "v2",
  "published": "2010-03-10T22:35:57.000Z",
  "updated": "2010-03-13T07:27:07.000Z",
  "title": "Transcendence of generating functions whose coefficients are multiplicative",
  "authors": [
    "Jason P. Bell",
    "Nils Bruin",
    "Michael Coons"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we give a new proof and an extension of the following result of B\\'ezivin. Let $f:\\B{N}\\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\\sum_{n\\geq 1} f(n)z^n\\in K[[z]]$ for its generating series. Suppose that $F(z)$ is 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, $F(z)$ is either transcendental or rational. For $K=\\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating series of a multiplicative function, then $F(z)$ is either transcendental or rational.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-03-13T07:27:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generating functions",
      "transcendence",
      "coefficients",
      "natural number",
      "finite generating series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.2221B"
    }
  }
}