{
  "id": "math/0411498",
  "version": "v1",
  "published": "2004-11-22T21:10:03.000Z",
  "updated": "2004-11-22T21:10:03.000Z",
  "title": "The structure of Bernoulli numbers",
  "authors": [
    "Bernd C. Kellner"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form $$ B_n = (-1)^{\\frac{n}{2}-1} \\prod_{p-1 \\nmid n} |n|_p^{-1} \\prod\\limits_{(p,l)\\in\\Psi^{\\rm irr}_1 \\atop n \\equiv l \\mods{p-1}} |p (\\chi_{(p,l)} - {\\textstyle \\frac{n-l}{p-1}})|_p^{-1} \\prod\\limits_{p-1 \\mid n} p^{-1} $$ where the $\\chi_{(p,l)}$ are zeros of certain $p$-adic zeta functions and $\\Psi^{\\rm irr}_1$ is the set of irregular pairs. The more complicated but improbable case where the conjecture does not hold is also handled; we obtain an unconditional structural formula for Bernoulli numbers. Finally, applications are given which are related to classical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-11-22T21:10:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68"
    ],
    "keywords": [
      "bernoulli numbers",
      "adic zeta functions",
      "unconditional structural formula",
      "conjecture",
      "irregular pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11498K"
    }
  }
}