{
  "id": "2103.06853",
  "version": "v1",
  "published": "2021-03-11T18:29:24.000Z",
  "updated": "2021-03-11T18:29:24.000Z",
  "title": "Expansion, divisibility and parity",
  "authors": [
    "Harald Andrés Helfgott",
    "Maksym Radziwiłł"
  ],
  "comment": "98 pages, 3 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbf{P} \\subset [H_0,H]$ be a set of primes, where $\\log H_0 \\geq (\\log H)^{2/3 + \\epsilon}$. Let $\\mathscr{L} = \\sum_{p \\in \\mathbf{P}} 1/p$. Let $N$ be such that $\\log H \\leq (\\log N)^{1/2-\\epsilon}$. We show there exists a subset $\\mathscr{X} \\subset (N, 2N]$ of density close to $1$ such that all the eigenvalues of the linear operator $$(A_{|\\mathscr{X}} f)(n) = \\sum_{\\substack{p \\in \\mathbf{P} : p | n \\\\ n, n \\pm p \\in \\mathscr{X}}} f(n \\pm p) \\; - \\sum_{\\substack{p \\in\\mathbf{P} \\\\ n, n \\pm p \\in \\mathscr{X}}} \\frac{f(n \\pm p)}{p}$$ are $O(\\sqrt{\\mathscr{L}})$. This bound is optimal up to a constant factor. In other words, we prove that a graph describing divisibility by primes is a strong local expander almost everywhere, and indeed within a constant factor of being \"locally Ramanujan\" (a.e.). Specializing to $f(n) = \\lambda(n)$ with $\\lambda(n)$ the Liouville function, and using an estimate by Matom\\\"aki, Radziwi{\\l}{\\l} and Tao on the average of $\\lambda(n)$ in short intervals, we derive that \\[\\frac{1}{\\log x} \\sum_{n\\leq x} \\frac{\\lambda(n) \\lambda(n+1)}{n} = O\\Big(\\frac{1}{\\sqrt{\\log \\log x}}\\Big),\\] improving on a result of Tao's. We also prove that $\\sum_{N<n\\leq 2 N} \\lambda(n) \\lambda(n+1)=o(N)$ at almost all scales with a similar error term, improving on a result by Tao and Ter\\\"av\\\"ainen. (Tao and Tao-Ter\\\"av\\\"ainen followed a different approach, based on entropy, not expansion; significantly, we can take a much larger value of $H$, and thus consider many more primes.) We can also prove sharper results with ease. For instance: let $S_{N,k}$ the set of all $N<n\\leq 2N$ such that $\\Omega(n) = k$. Then, for any fixed value of $k$ with $k = \\log \\log N + O(\\sqrt{\\log \\log N})$ (that is, any \"popular\" value of $k$) the average of $\\lambda(n+1)$ over $S_{N,k}$ is $o(1)$ at almost all scales.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-11T18:29:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C48",
      "11N37",
      "11N35",
      "05C81"
    ],
    "keywords": [
      "divisibility",
      "constant factor",
      "strong local expander",
      "similar error term",
      "liouville function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 98,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}