{
  "id": "1711.02871",
  "version": "v1",
  "published": "2017-11-08T08:43:53.000Z",
  "updated": "2017-11-08T08:43:53.000Z",
  "title": "On the Number of Connected Components of Ranges of Divisor Functions",
  "authors": [
    "Nina Zubrilina"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For $r \\in \\mathbb{R}, r> 1$ and $n \\in \\mathbb{Z}^+$, the divisor function $\\sigma_{-r}$ is defined by $\\sigma_{-r}(n) := \\sum_{d \\vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of $\\overline{\\sigma_{-r}(\\mathbb{Z}^+)}$ satisfies $$\\pi(r) + 1 \\leq C_r \\leq \\frac{1}{2}\\exp\\left[\\frac{1}{2}\\dfrac{r^{20/9}}{(\\log r)^{29/9}} \\left( 1 + \\frac{\\log\\log r}{\\log r - \\log\\log r} + \\frac{\\mathcal{O}(1)}{\\log r}\\right) \\right],$$ where $\\pi(t)$ is the number of primes $p\\leq t$. We also show that $C_r$ does not take all integer values, specifically that it cannot be equal to $4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-08T08:43:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25"
    ],
    "keywords": [
      "divisor function",
      "connected components",
      "integer values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}