{
  "id": "1511.02558",
  "version": "v1",
  "published": "2015-11-09T03:31:21.000Z",
  "updated": "2015-11-09T03:31:21.000Z",
  "title": "The Log-Behavior of $\\sqrt[n]{p(n)}$ and $\\sqrt[n]{p(n)/n}$",
  "authors": [
    "William Y. C. Chen",
    "Ken Y. Zheng"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\\frac{p(n-1)}{p(n)}\\left(1+\\frac{1}{n}\\right)>\\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\\sqrt[n]{p(n)/n}$ and $\\Delta$ be the difference operator respect to $n$. Desalvo and Pak pointed out that their approach to proving the log-concavity of $p(n)$ may be employed to prove a conjecture of Sun on the log-convexity of $\\{r(n)\\}_{n\\geq 61}$, as long as one finds an appropriate estimate of $\\Delta^2 \\log r(n-1)$. In this paper, we obtain a lower bound for $\\Delta^2\\log r(n-1)$, leading to a proof of this conjecture. From the log-convexity of $\\{r(n)\\}_{n\\geq61}$ and $\\{\\sqrt[n]{n}\\}_{n\\geq4}$, we are led to a proof of another conjecture of Sun on the log-convexity of $\\{\\sqrt[n]{p(n)}\\}_{n\\geq27}$. Furthermore, we show that $\\lim\\limits_{n \\rightarrow +\\infty}n^{\\frac{5}{2}}\\Delta^2\\log\\sqrt[n]{p(n)}=3\\pi/\\sqrt{24}$. Finally, by finding an upper bound of $\\Delta^2 \\log\\sqrt[n-1]{p(n-1)}$, we prove an inequality on the ratio $\\frac{\\sqrt[n-1]{p(n-1)}}{\\sqrt[n]{p(n)}}$ analogous to the above inequality on the ratio $\\frac{p(n-1)}{p(n)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-09T03:31:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20"
    ],
    "keywords": [
      "log-behavior",
      "conjecture",
      "log-convexity",
      "difference operator respect",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}