{
  "id": "1606.05722",
  "version": "v1",
  "published": "2016-06-18T06:21:03.000Z",
  "updated": "2016-06-18T06:21:03.000Z",
  "title": "Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers",
  "authors": [
    "Khodabakhsh Hessami Pilehrood",
    "Tatiana Hessami Pilehrood",
    "Roberto Tauraso"
  ],
  "comment": "Submitted on January 14, 2016",
  "categories": [
    "math.NT"
  ],
  "abstract": "It is well known that the harmonic sum $H_n(1)=\\sum_{k=1}^n\\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erd\\H{o}s and Niven proved that the nested multiple harmonic sum $H_n(\\{1\\}^r)=\\sum_{1\\le k_1<\\dots<k_r\\le n}\\frac{1}{k_1\\cdots k_r}$ can take integer values only for a finite number of positive integers $n$. In 2012, Chen and Tang refined this result by showing that $H_n(\\{1\\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider the integrality problem for arbitrary multiple harmonic and multiple harmonic star sums and show that none of these sums is an integer with some natural exceptions like those mentioned above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-18T06:21:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M32",
      "11N05",
      "11Y70",
      "11B75"
    ],
    "keywords": [
      "multiple harmonic star sums",
      "nested multiple harmonic sum",
      "arbitrary multiple harmonic",
      "integer values",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}