{
  "id": "2012.04914",
  "version": "v1",
  "published": "2020-12-09T08:34:26.000Z",
  "updated": "2020-12-09T08:34:26.000Z",
  "title": "On the least common multiple of random $q$-integers",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For every positive integer $n$ and for every $\\alpha \\in [0, 1]$, let $\\mathcal{B}(n, \\alpha)$ denote the probabilistic model in which a random set $\\mathcal{A} \\subseteq \\{1, \\dots, n\\}$ is constructed by picking independently each element of $\\{1, \\dots, n\\}$ with probability $\\alpha$. Cilleruelo, Ru\\'{e}, \\v{S}arka, and Zumalac\\'{a}rregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $\\mathcal{A}$. Let $q$ be an indeterminate and let $[k]_q := 1 + q + q^2 + \\cdots + q^{k-1} \\in \\mathbb{Z}[q]$ be the $q$-analog of the positive integer $k$. We determine the expected value and the variance of $X := \\operatorname{deg} \\operatorname{lcm}\\!\\big([\\mathcal{A}]_q\\big)$, where $[\\mathcal{A}]_q := \\big\\{[k]_q : k \\in \\mathcal{A}\\big\\}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-09T08:34:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11B99"
    ],
    "keywords": [
      "common multiple",
      "sure asymptotic formula",
      "positive integer",
      "cilleruelo",
      "random set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}