{
  "id": "0809.1691",
  "version": "v1",
  "published": "2008-09-09T23:52:55.000Z",
  "updated": "2008-09-09T23:52:55.000Z",
  "title": "Completely multiplicative functions taking values in $\\{-1,1\\}$",
  "authors": [
    "Peter Borwein",
    "Stephen K. K. Choi",
    "Michael Coons"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Define {\\em the Liouville function for $A$}, a subset of the primes $P$, by $\\lambda_{A}(n) =(-1)^{\\Omega_A(n)}$ where $\\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\\sum_{k\\leq n}\\lambda_A(n)\\quad{and}\\quad R_A:=\\lim_{n\\to\\infty}\\frac{L_A(n)}{n}.$$ We show that for every $\\alpha\\in[0,1]$ there is an $A\\subset P$ such that $R_A=\\alpha$. Given certain restrictions on $A$, asymptotic estimates for $\\sum_{k\\leq n}\\lambda_A(k)$ are also given. With further restrictions, more can be said. For {\\em character--like functions} $\\lambda_p$ ($\\lambda_p$ agrees with a Dirichlet character $\\chi$ when $\\chi(n)\\neq 0$) exact values and asymptotics are given; in particular $$\\quad\\sum_{k\\leq n}\\lambda_p(k)\\ll \\log n.$$ Within the course of discussion, the ratio $\\phi(n)/\\sigma(n)$ is considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-09T23:52:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "11N37"
    ],
    "keywords": [
      "multiplicative functions",
      "traditional liouville function",
      "exact values",
      "dirichlet character",
      "asymptotic estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.1691B"
    }
  }
}