{
  "id": "1307.1414",
  "version": "v1",
  "published": "2013-07-04T17:06:45.000Z",
  "updated": "2013-07-04T17:06:45.000Z",
  "title": "On the average number of subgroups of the group $\\Z_m \\times \\Z_n$",
  "authors": [
    "Werner Georg Nowak",
    "László Tóth"
  ],
  "comment": "14 pages",
  "journal": "Int. J. Number Theory 10 (2014), 363-374",
  "doi": "10.1142/S179304211350098X",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "Let $\\Z_m$ be the group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups of the group $\\Z_m \\times \\Z_n$ and the number of its cyclic subgroups, respectively, where $m$ and $n$ are arbitrary positive integers. We derive asymptotic formulas for the sums $\\sum_{m,n\\le x} s(m,n)$, $\\sum_{m,n\\le x} c(m,n)$ and for the corresponding sums restricted to $\\gcd(m,n)>1$, i.e., concerning the groups $\\Z_m \\times \\Z_n$ having rank two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-04T17:06:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37",
      "20K01",
      "20K27"
    ],
    "keywords": [
      "average number",
      "residue classes modulo",
      "cyclic subgroups",
      "arbitrary positive integers",
      "derive asymptotic formulas"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.1414N"
    }
  }
}