{
  "id": "1808.07550",
  "version": "v1",
  "published": "2018-08-22T20:11:37.000Z",
  "updated": "2018-08-22T20:11:37.000Z",
  "title": "On Gaps in the Closures of Images of Divisor Functions",
  "authors": [
    "Niven Achenjang",
    "Aaron Berger"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a complex number $c$, define the divisor function $\\sigma_c:\\mathbb N\\to\\mathbb C$ by $\\sigma_c(n)=\\sum_{d\\mid n}d^c$. In this paper, we look at $\\overline{\\sigma_{-r}(\\mathbb N)}$, the topological closures of the image of $\\sigma_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\\overline{\\sigma_{-r}(\\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\\overline{\\sigma_{-r}(\\mathbb N)}$ in order to work towards a possible monotonicity result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-22T20:11:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N64"
    ],
    "keywords": [
      "divisor function",
      "general structure",
      "lower bounds",
      "monotonicity result",
      "complex number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}