{
  "id": "2402.03999",
  "version": "v1",
  "published": "2024-02-06T13:52:46.000Z",
  "updated": "2024-02-06T13:52:46.000Z",
  "title": "An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field",
  "authors": [
    "Ilaria Viglino"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For $ f\\in\\mathbb{Z}[X] $ an irreducible polynomial of degree $ n $, the Cilleruelo's conjecture states that$$\\log(\\mbox{lcm}(f(1),\\dots,f(M)))\\sim(n-1)M\\log M$$as $ M\\rightarrow+\\infty $, where $ \\mbox{lcm}(f(1),\\dots,f(M)) $ is the least common multiple of $f(1),\\dots,f(M)$. It's well-known for $ n=1 $ as a consequence of Dirichlet's Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for $S_n$-polynomials with integral coefficients over a fixed extension $K/\\mathbb{Q}$ by considering the least common multiple of ideals of $\\mathcal{O}_K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-06T13:52:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "average version",
      "number field",
      "common multiple",
      "cilleruelos conjecture states",
      "dirichlets theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}