{
  "id": "2102.12297",
  "version": "v1",
  "published": "2021-02-24T14:21:11.000Z",
  "updated": "2021-02-24T14:21:11.000Z",
  "title": "Correlations of Almost Primes",
  "authors": [
    "Natalie Evans"
  ],
  "comment": "34 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_1,\\ldots,p_5$ be primes with $p_1p_2,p_5\\in(X,2X]$, let $h$ be a non-zero integer and $A > 3$. Supposing that $p_1,p_3$ lie in a certain range depending on $X$, we prove an asymptotic for the weighted number of solutions to $p_3p_4=p_1p_2+h$ which holds for almost all $0 < |h|\\leq H$ with $\\log^{19+\\varepsilon}X\\leq H\\leq X\\log^{-A}X$. Using the same methods and supposing instead that $p_1p_2, p_3p_4$ have the typical factorisation we prove an asymptotic which holds for almost all $0 < |h|\\leq H$ with $\\exp\\left((\\log X)^{1-o(1)}\\right)\\leq H\\leq X\\log^{-A}X$ and an asymptotic for the weighted number of solutions to $p_3p_4=p_5+h$ which holds for almost all $0 < |h|\\leq H$ with $X^{1/6+\\varepsilon}\\leq H\\leq X\\log^{-B}X$ and $B > 5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-24T14:21:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "correlations",
      "weighted number",
      "asymptotic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}