{
  "id": "1406.3698",
  "version": "v2",
  "published": "2014-06-14T07:53:12.000Z",
  "updated": "2016-01-16T18:25:55.000Z",
  "title": "On an asymptotic behavior of the divisor function $τ(n)$",
  "authors": [
    "Tigran Hakobyan"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For $\\mu>0$ we study an asymptotic behavior of the sequence defined as $$T_{n}(\\mu)=\\frac{max_{1\\leq m \\leq {n^{\\frac{1}{\\mu}}}}\\{\\tau (n + m)\\}}{\\tau(n)},\\ n=1,2,...$$ where $\\tau(n)$ denotes the number of natural divisors of the given $n\\in \\mathbb{N}$. The motivation of this observation is to explore whether $\\tau$ function oscillates rapidly in small neighborhoods of natural numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-14T07:53:12.000Z",
      "abstract": "In this paper we study an asymptotic behavior of the sequence defined as $T_{n}(\\mu)=\\frac{max_{1\\leq m \\leq \\sqrt[\\mu]{n}}\\{\\tau (n + m)\\}}{\\tau(n)}$ where $\\mu>0$ is fixed and $\\tau(n)$ denotes the number of natural divisors of the given $n\\in N$.",
      "comment": "10 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-01-16T18:25:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "divisor function",
      "natural divisors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3698H"
    }
  }
}