{
  "id": "0911.4433",
  "version": "v8",
  "published": "2009-11-23T19:27:44.000Z",
  "updated": "2013-10-30T15:02:30.000Z",
  "title": "Arithmetic theory of harmonic numbers (II)",
  "authors": [
    "Zhi-Wei Sun",
    "Li-Lu Zhao"
  ],
  "comment": "13 pages. Final published version",
  "journal": "Colloq. Math. 130(2013), 67-78",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For $k=1,2,\\ldots$ let $H_k$ denote the harmonic number $\\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have $$\\sum_{k=1}^{p-1}\\frac{H_k}{k2^k}\\equiv\\frac7{24}pB_{p-3}\\pmod{p^2},\\ \\ \\sum_{k=1}^{p-1}\\frac{H_{k,2}}{k2^k}\\equiv-\\frac 38B_{p-3}\\pmod{p},$$ and $$\\sum_{k=1}^{p-1}\\frac{H_{k,2n}^2}{k^{2n}}\\equiv\\frac{\\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n}\\pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers, and $H_{k,m}:=\\sum_{j=1}^k 1/j^m$.",
  "revisions": [
    {
      "version": "v8",
      "updated": "2013-10-30T15:02:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B68",
      "05A19",
      "11B75"
    ],
    "keywords": [
      "harmonic number",
      "arithmetic theory",
      "bernoulli numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.4433S"
    }
  }
}