{
  "id": "2004.07335",
  "version": "v1",
  "published": "2020-04-15T20:50:13.000Z",
  "updated": "2020-04-15T20:50:13.000Z",
  "title": "Nontrivial upper bounds for the least common multiple of an arithmetic progression",
  "authors": [
    "Sid Ali Bousla"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with $b\\geq 2$, we have \\[\\mathrm{lcm}\\left(a,a+b,\\dots,a+nb\\right) \\leq \\left(c_1\\cdot b\\log b\\right)^{n+\\left\\lfloor \\frac{a}{b}\\right\\rfloor}~~~~(\\forall n\\geq b+1),\\] where $c_1=41.30142$. If in addition $b$ is a prime number and $a<b$, then we prove that for any $n\\geq b+1$, we have $\\mathrm{lcm}\\left(a,a+b,\\dots,a+nb\\right) \\leq \\left(c_2\\cdot b^{\\frac{b}{b-1}}\\right)^n$, where $c_2=12.30641$. Finally, we apply those inequalities to estimate the arithmetic function $M$ defined by $M(n):=\\frac{1}{\\varphi(n)}\\sum_{\\substack{1\\leq\\ell\\leq n \\\\ \\ell \\wedge n=1}}\\frac{1}{\\ell}$ ($\\forall n \\geq 1$), as well as some values of the generalized Chebyshev function $\\theta(x;k,\\ell)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-15T20:50:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11B25"
    ],
    "keywords": [
      "nontrivial upper bounds",
      "common multiple",
      "finite arithmetic progression",
      "effective upper bounds",
      "arithmetic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}