{
  "id": "1604.04508",
  "version": "v1",
  "published": "2016-04-15T13:53:26.000Z",
  "updated": "2016-04-15T13:53:26.000Z",
  "title": "On the average value of the least common multiple of $k$ positive integers",
  "authors": [
    "Titus Hilberdink",
    "László Tóth"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We deduce an asymptotic formula with error term for the sum $\\sum_{n_1,\\ldots,n_k \\le x} f([n_1,\\ldots, n_k])$, where $[n_1,\\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\\ldots, n_k$ ($k\\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $\\varphi(n)^r$, $\\sigma(n)^r$ ($r>-1$ real), where $\\varphi$ is Euler's totient function and $\\sigma$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function $f$, the function of $k$ variables $f([n_1,\\ldots,n_k])$ is multiplicative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-15T13:53:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A25",
      "11N37"
    ],
    "keywords": [
      "common multiple",
      "positive integers",
      "average value",
      "eulers totient function",
      "multiplicative arithmetic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160404508H"
    }
  }
}