{
  "id": "1304.3922",
  "version": "v2",
  "published": "2013-04-14T16:28:33.000Z",
  "updated": "2013-07-02T15:44:48.000Z",
  "title": "Secant Zeta Functions",
  "authors": [
    "Matilde Lalín",
    "Francis Rodrigue",
    "Mathew Rogers"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the series $\\psi_s(z):=\\sum_{n=1}^{\\infty} \\sec(n\\pi z)n^{-s}$, and prove that it converges under mild restrictions on $z$ and $s$. The function possesses a modular transformation property, which allows us to evaluate $\\psi_{s}(z)$ explicitly at certain quadratic irrational values of $z$. This supports our conjecture that $\\pi^{-k} \\psi_{k}(\\sqrt{j})\\in\\mathbb{Q}$ whenever $k$ and $j$ are positive integers with $k$ even. We conclude with some speculations on Bernoulli numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-02T15:44:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33E20",
      "33B30",
      "11L03"
    ],
    "keywords": [
      "secant zeta functions",
      "quadratic irrational values",
      "modular transformation property",
      "mild restrictions",
      "bernoulli numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.3922L"
    }
  }
}