{
  "id": "2111.08912",
  "version": "v2",
  "published": "2021-11-17T05:33:08.000Z",
  "updated": "2022-05-30T15:21:32.000Z",
  "title": "On the Hardy-Littlewood-Chowla conjecture on average",
  "authors": [
    "Jared Duker Lichtman",
    "Joni Teräväinen"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any $k,\\ell\\ge1$ and distinct integers $h_2,\\ldots,h_k,a_1,\\ldots,a_\\ell$, we have $$\\sum_{n\\leq X}\\mu(n+h_1)\\cdots \\mu(n+h_k)\\Lambda(n+a_1)\\cdots\\Lambda(n+a_{\\ell})=o(X)$$ for all except $o(H)$ values of $h_1\\leq H$, so long as $H\\geq (\\log X)^{\\ell+\\varepsilon}$. This improves on the range $H\\ge (\\log X)^{\\psi(X)}$, $\\psi(X)\\to\\infty$, obtained in previous work of the first author. Our results also generalize from the M\\\"obius function $\\mu$ to arbitrary (non-pretentious) multiplicative functions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-05-30T15:21:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P32",
      "11L20",
      "11N37"
    ],
    "keywords": [
      "hardy-littlewood-chowla conjecture",
      "first author",
      "hybrid form",
      "distinct integers",
      "celebrated conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}