{
  "id": "1209.4329",
  "version": "v2",
  "published": "2012-09-19T18:54:36.000Z",
  "updated": "2014-10-28T18:32:15.000Z",
  "title": "On quotients of Riemann zeta values at odd and even integer arguments",
  "authors": [
    "Bernd C. Kellner"
  ],
  "comment": "14 pages; final revised version",
  "journal": "J. Number Theory 133 (2013), No. 8, 2684-2698",
  "doi": "10.1016/j.jnt.2013.02.008",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show for even positive integers $n$ that the quotient of the Riemann zeta values $\\zeta(n+1)$ and $\\zeta(n)$ satisfies the equation $$\\frac{\\zeta(n+1)}{\\zeta(n)} = (1-\\frac{1}{n}) (1-\\frac{1}{2^{n+1}-1}) \\frac{\\mathcal{L}^\\star(\\mathfrak{p}_n)}{\\mathfrak{p}_n'(0)},$$ where $\\mathfrak{p}_n \\in \\mathbb{Z}[x]$ is a certain monic polynomial of degree $n$ and $\\mathcal{L}^\\star: \\mathbb{C}[x] \\to \\mathbb{C}$ is a linear functional, which is connected with a special $L$-function. There exists the decomposition $\\mathfrak{p}_n(x) = x(x+1) \\mathfrak{q}_n(x)$. If $n = p+1$ where $p$ is an odd prime, then $\\mathfrak{q}_n$ is an Eisenstein polynomial and therefore irreducible over $\\mathbb{Z}[x]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-19T18:54:36.000Z",
      "abstract": "We show for even positive integers $n$ that the quotient of the Riemann zeta values $\\zeta(n+1)$ and $\\zeta(n)$ satisfies the equation [\\frac{\\zeta(n+1)}{\\zeta(n)} = (1-\\frac{1}{n}) (1-\\frac{1}{2^{n+1}-1}) \\frac{\\mathcal{L}^\\star(\\mathfrak{p}_n)}{\\mathfrak{p}_n'(0)},] where $\\mathfrak{p}_n \\in \\ZZ[x]$ is a certain monic polynomial of degree $n$ and $\\mathcal{L}^\\star: \\CC[x] \\to \\CC$ is a linear functional, which is connected with a special $L$-function. There exists the decomposition $\\mathfrak{p}_n(x) = x(x+1) \\mathfrak{q}_n(x)$. If $n = p+1$ where $p$ is an odd prime, then $\\mathfrak{q}_n$ is an Eisenstein polynomial and therefore irreducible over $\\ZZ[x]$.",
      "comment": "14 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-28T18:32:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11R09",
      "11B73"
    ],
    "keywords": [
      "riemann zeta values",
      "integer arguments",
      "eisenstein polynomial",
      "linear functional",
      "odd prime"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.4329K"
    }
  }
}