{
  "id": "1911.00168",
  "version": "v1",
  "published": "2019-11-01T00:47:17.000Z",
  "updated": "2019-11-01T00:47:17.000Z",
  "title": "An improved bound on the least common multiple of polynomial sequences",
  "authors": [
    "Ashwin Sah"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Cilleruelo conjectured that if $f\\in\\mathbb{Z}[x]$ of degree $d$ is irreducible over the rationals, then $\\log\\operatorname{lcm}(f(1),\\ldots,f(N))\\sim(d-1)N\\log N$ as $N\\to\\infty$. He proved it for the case $d = 2$. Very recently, Maynard and Rudnick proved there exists $c_d > 0$ with $\\log\\operatorname{lcm}(f(1),\\ldots,f(N))\\gtrsim c_d N\\log N$, and showed one can take $c_d = \\frac{d-1}{d^2}$. We give an alternative proof of this result with the improved constant $c_d = \\frac{2}{d}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-01T00:47:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "common multiple",
      "polynomial sequences",
      "cilleruelo",
      "alternative proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}