{
  "id": "2008.07850",
  "version": "v1",
  "published": "2020-08-18T10:58:16.000Z",
  "updated": "2020-08-18T10:58:16.000Z",
  "title": "On the weighted average number of subgroups of ${\\mathbb {Z}}_{m}\\times {\\mathbb {Z}}_{n}$ with $mn\\leq x$",
  "authors": [
    "Isao Kiuchi",
    "Sumaia Saad Eddin"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\\mathbb{Z}}_{m}\\times {\\mathbb{Z}}_{n}$, respectively. Define $$ \\widetilde{D}_{s}(x) = \\sum_{mn\\leq x}s(m,n)\\log\\frac{x}{mn} \\quad \\quad \\widetilde{D}_{c}(x) = \\sum_{mn\\leq x}c(m,n)\\log\\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\\widetilde{D}_{s}(x)$ and $\\widetilde{D}_{c}(x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T10:58:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37",
      "11Y60"
    ],
    "keywords": [
      "weighted average number",
      "residue classes modulo",
      "asymptotic behaviour",
      "total number",
      "cyclic subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}