{
  "id": "1407.0177",
  "version": "v1",
  "published": "2014-07-01T10:37:52.000Z",
  "updated": "2014-07-01T10:37:52.000Z",
  "title": "Finite Differences of the Logarithm of the Partition Function",
  "authors": [
    "William Y. C. Chen",
    "Larry X. W. Wang",
    "Gary Y. B. Xie"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\\frac{p(n-1)}{p(n)}\\left(1+\\frac{1}{n}\\right)> \\frac{p(n)}{p(n+1)}$ for $n\\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\\frac{p(n-1)}{p(n)}\\left( 1+\\frac{\\pi}{\\sqrt{24}n^{3/2}}\\right) > \\frac{p(n)}{p(n+1)}$ holds for $n\\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\\Delta^{2} \\log p(n-1)$, where $\\Delta$ is the difference operator with respect to $n$. We also show that for given $r\\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\\Delta^{r} \\log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $r\\geq 1$, $\\Delta^{r} p(n)>0$ for sufficiently large $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-01T10:37:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A20",
      "11B68"
    ],
    "keywords": [
      "partition function",
      "finite differences",
      "conjecture",
      "sufficiently large",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0177C"
    }
  }
}