{
  "id": "2312.06012",
  "version": "v1",
  "published": "2023-12-10T21:52:46.000Z",
  "updated": "2023-12-10T21:52:46.000Z",
  "title": "On Correlations of Liouville-like Functions",
  "authors": [
    "Yichen You"
  ],
  "comment": "9 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathcal{A}$ be a set of mutually coprime positive integers, satisfying \\begin{align*} \\sum\\limits_{a\\in\\mathcal{A}}\\frac{1}{a} = \\infty. \\end{align*} Define the (possibly non-multiplicative) \"Liouville-like\" functions \\begin{align*} \\lambda_{\\mathcal{A}}(n) = (-1)^{\\#\\{a:a|n, a \\in \\mathcal{A}\\}} \\text{ or } (-1)^{\\#\\{a:a^\\nu\\parallel n, a \\in \\mathcal{A}, \\nu \\in \\mathbb{N}\\}}. \\end{align*} We show that \\begin{align*} \\lim\\limits_{x\\to\\infty}\\frac{1}{x}\\sum\\limits_{n \\leq x} \\lambda_\\mathcal{A}(n) = 0 \\end{align*} holds, answering a question of de la Rue.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-10T21:52:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville-like functions",
      "correlations",
      "mutually coprime positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}