{
  "id": "1904.05096",
  "version": "v1",
  "published": "2019-04-10T10:22:53.000Z",
  "updated": "2019-04-10T10:22:53.000Z",
  "title": "Value patterns of multiplicative functions and related sequences",
  "authors": [
    "Terence Tao",
    "Joni Teräväinen"
  ],
  "comment": "42 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short intervals in a suitable sense, we show that the asymptotic density of the pattern $n+1\\in A$, $n+2\\in A$, $n+3\\in A$ is positive, as long as $A$ has density greater than $\\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\\frac{1}{3}$, below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results firstly to answer in a stronger form a question of Erd\\H{o}s and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$, $P^{+}(n+1), P^{+}(n+2)$ of three consecutive integers. Secondly, we show that the tuple $(\\omega(n+1),\\omega(n+2),\\omega(n+3)) \\pmod 3$ takes all the $27$ possible patterns in $(\\mathbb{Z}/3\\mathbb{Z})^3$ with positive lower density, with $\\omega(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+i\\in A_i$, $i=1,\\dots k$ in approximately multiplicative sets $A_i$ having large enough densities, generalising some results of Hildebrand on his 'stable sets conjecture'. Lastly, we consider the sign patterns of the Liouville function $\\lambda$ and show that there are at least $24$ patterns of length $5$ that occur with positive density. In all of the proofs we make extensive use of recent ideas concerning correlations of multiplicative functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-10T10:22:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "37A45"
    ],
    "keywords": [
      "multiplicative functions",
      "value patterns",
      "related sequences",
      "largest prime factors",
      "distinct prime divisors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}