{
  "id": "1208.5213",
  "version": "v3",
  "published": "2012-08-26T10:37:36.000Z",
  "updated": "2013-09-27T02:55:00.000Z",
  "title": "Zeta Functions and the Log-behavior of Combinatorial Sequences",
  "authors": [
    "William Y. C. Chen",
    "Jeremy J. F. Guo",
    "Larry X. W. Wang"
  ],
  "comment": "16 pages; to appear in Proc. Edinburgh Math. Soc. (2)",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we use the Riemann zeta function $\\zeta(x)$ and the Bessel zeta function $\\zeta_{\\mu}(x)$ to study the log-behavior of combinatorial sequences. We prove that $\\zeta(x)$ is log-convex for $x>1$. As a consequence, we deduce that the sequence $\\{|B_{2n}|/(2n)!\\}_{n\\geq 1}$ is log-convex, where $B_n$ is the $n$-th Bernoulli number. We introduce the function $\\theta(x)=(2\\zeta(x)\\Gamma(x+1))^{\\frac{1}{x}}$, where $\\Gamma(x)$ is the gamma function, and we show that $\\log \\theta(x)$ is strictly increasing for $x\\geq 6$. This confirms a conjecture of Sun stating that the sequence $\\{\\sqrt[n] {|B_{2n}}|\\}_{n\\geq 1}$ is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers $a_n(\\mu)=2^{2n+1}(n+1)!(\\mu+1)_n\\zeta_{\\mu}(2n)$ and conjectured that the sequence $\\{a_n(\\mu)\\}_{n\\geq 1}$ is log-convex for $\\mu=0$ and $\\mu=1$. By proving that $\\zeta_{\\mu}(x)$ is log-convex for $x>1$ and $\\mu>-1$, we show that the sequence $\\{a_n(\\mu)\\}_{n\\geq 1}$ is log-convex for any $\\mu>-1$. We introduce another function $\\theta_{\\mu}(x)$ involving $\\zeta_{\\mu}(x)$ and the gamma function $\\Gamma(x)$ and we show that $\\log \\theta_{\\mu}(x)$ is strictly increasing for $x>8e(\\mu+2)^2$. This implies that $\\sqrt[n]{a_n(\\mu)}<\\sqrt[n+1]{a_{n+1}(\\mu)}$ for $n> 4e(\\mu+2)^2$. Based on Dobinski's formula, we prove that $\\sqrt[n]{B_n}<\\sqrt[n+1]{B_{n+1}}$ for $n\\geq 1$, where $B_n$ is the $n$-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of $\\{\\sqrt[n]{B_n}\\}_{n\\geq 1}$ and H\\\"{o}lder's inequality in probability theory.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-27T02:55:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "11B68"
    ],
    "keywords": [
      "combinatorial sequences",
      "log-behavior",
      "log-convex",
      "gamma function",
      "riemann zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5213C"
    }
  }
}