{
  "id": "2308.15146",
  "version": "v1",
  "published": "2023-08-29T09:27:20.000Z",
  "updated": "2023-08-29T09:27:20.000Z",
  "title": "Square-free values of polynomials on average",
  "authors": [
    "Pascal Jelinek"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was able to show conditionally on the $abc$-conjecture that this conjecture is true for polynomials of arbitrarily large degrees. In 2013 Shparlinski proved that this conjecture holds on average over all polynomials of a fixed naive height, which was improved by Browning and Shparlinski in 2023. In this paper, we improve the dependence between $x$ and the height of the polynomial. We achieve this via adapting a method introduced in a 2022 paper by Browning, Sofos, and Ter\\\"av\\\"ainen on the Bateman-Horn conjecture, the polynomial Chowla conjecture, and the Hasse principle on average.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-29T09:27:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N32"
    ],
    "keywords": [
      "square-free values",
      "polynomial chowla conjecture",
      "shparlinski",
      "hasse principle",
      "bateman-horn conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}