{
  "id": "2109.06291",
  "version": "v1",
  "published": "2021-09-13T20:00:59.000Z",
  "updated": "2021-09-13T20:00:59.000Z",
  "title": "The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero",
  "authors": [
    "Terence Tao",
    "Joni Teräväinen"
  ],
  "comment": "53 pages, no figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\\dots,h_k,h'_1,\\dots,h'_\\ell$, we establish an asymptotic formula for $$\\sum_{n\\leq x}\\Lambda(n+h_1)\\cdots \\Lambda(n+h_k)\\lambda(n+h_{1}')\\cdots \\lambda(n+h_{\\ell}')$$ for any $0\\leq k\\leq 2$ and $\\ell \\geq 0$. Specializing to either $\\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T20:00:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N36"
    ],
    "keywords": [
      "siegel zero",
      "hardy-littlewood-chowla conjecture",
      "hardy-littlewood prime tuples conjectures",
      "asymptotic formula",
      "infinite sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}