{
  "id": "1704.08358",
  "version": "v1",
  "published": "2017-04-26T21:35:58.000Z",
  "updated": "2017-04-26T21:35:58.000Z",
  "title": "On the non-vanishing of certain Dirichlet series",
  "authors": [
    "Sandro Bettin",
    "Bruno Martin"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given $k\\in\\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\\sum_{n\\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and $p\\equiv 3\\mod 4$, then there are no odd rational-valued functions $f\\not\\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases there are examples of odd functions $f$ such that $D_k(1,f)=0$. As a consequence, we obtain, for example, that the set of values $L(1,\\chi)^2$, where $\\chi$ ranges over odd characters mod $p$, are linearly independent over $\\mathbb Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-26T21:35:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "11L03",
      "11M20",
      "11R18"
    ],
    "keywords": [
      "dirichlet series",
      "periodic function modulo",
      "odd characters mod",
      "non-vanishing",
      "odd rational-valued functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}