{
  "id": "1710.01195",
  "version": "v1",
  "published": "2017-10-03T14:54:11.000Z",
  "updated": "2017-10-03T14:54:11.000Z",
  "title": "On binary correlations of multiplicative functions",
  "authors": [
    "Joni Teräväinen"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study logarithmically averaged binary correlations of bounded multiplicative functions $g_1$ and $g_2$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_1$ or $g_2$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_j$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of $g_1$ and $g_2$ is asymptotic to the product of their mean values. We derive several applications, first showing that the number of large prime factors of $n$ and $n+1$ are independent of each other with respect to the logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erd\\H{o}s and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\\leq x^{4-\\varepsilon}$, the logarithmic average around $x$ of the real character $\\chi \\pmod{Q}$ over the values of a reducible quadratic polynomial is small.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-03T14:54:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N60",
      "11L40"
    ],
    "keywords": [
      "multiplicative functions",
      "study logarithmically averaged binary correlations",
      "large prime factors",
      "twisted dirichlet character",
      "wider class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}