{
  "id": "1608.04606",
  "version": "v1",
  "published": "2016-08-15T04:15:39.000Z",
  "updated": "2016-08-15T04:15:39.000Z",
  "title": "A recursive relation and some statistical properties for Möbius function",
  "authors": [
    "Rong Qiang Wei"
  ],
  "comment": "28 pages, 6 figues, 4 tables",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "An elementary recursive relation for M$\\ddot{\\mathrm{o}}$bius function $\\mu (n)$ is obtained by two ways. With this recursive relation, $\\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\\mu (1) \\sim \\mu (2 \\times 10^7) $ are calculated recursively one by one. Based on these $2\\times 10^7$ samples, the empirical probabilities of $\\mu (n)$ of taking $-1$, 0, and 1 in classic statistics are calculated and compared with the theoretical probabilities in number theory. The numerical consistency between these two kinds of probability show that $\\mu (n)$ could be seen as an independent random sequence when $n$ is large. The expectation and variance of the $\\mu (n)$ are $0$ and $6 n/ \\pi^2$, respectively. Furthermore, we show that any conjecture of the Mertens type is false in probability sense, and present an upper bound for cumulative sums of $\\mu (n)$ with a certain probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-15T04:15:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "möbius function",
      "statistical properties",
      "independent random sequence",
      "number theory",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}