{
  "id": "2010.07924",
  "version": "v1",
  "published": "2020-10-15T17:55:28.000Z",
  "updated": "2020-10-15T17:55:28.000Z",
  "title": "On the Liouville function at polynomial arguments",
  "authors": [
    "Joni Teräväinen"
  ],
  "comment": "43 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-S\\'ark\\\"ozy asks to show that if $P(x)\\in \\mathbb{Z}[x]$, then the sequence $\\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $\\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a \"99% version\" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T17:55:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11B30"
    ],
    "keywords": [
      "liouville function",
      "polynomial arguments",
      "changes sign",
      "multiplicative function analogue",
      "elliotts conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}