{
  "id": "1404.7221",
  "version": "v2",
  "published": "2014-04-29T03:13:02.000Z",
  "updated": "2015-06-02T11:36:39.000Z",
  "title": "Numerical calculation of the Riemann zeta function at odd integer arguments: A direct formula method",
  "authors": [
    "Qiang Luo",
    "Zhidan Wang"
  ],
  "comment": "12 pages, 1 figure",
  "journal": "Mathematical Sciences March 2015, Volume 9, Issue 1, pp 39-45",
  "doi": "10.1007/s40096-015-0146-9",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Based on the formula, an algorithm to evaluate $\\zeta$-values(i.e. the values of Riemann zeta function) at odd-integers from the two nearest $\\zeta$-values at even-integers is posed and proved. The behavior of the error bound is $O(10^{-n})$ approximately where $n$ is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones it can also reach spectacular accuracies such as more than ten decimal places.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-29T03:13:02.000Z",
      "abstract": "In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Basing on the formula, we construct an algorithm to evaluate $\\zeta$-values at odd-integers from the two nearest $\\zeta$-values at even-integers. The behavior of the error bound is $O(10^{-n})$ approximately where $n$ is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones it can also reach spectacular accuracies such as more than ten decimal places.",
      "comment": "9 pages, 0 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-02T11:36:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta function",
      "direct formula method",
      "odd integer arguments",
      "numerical calculation",
      "reach spectacular accuracies"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1293070,
      "adsabs": "2014arXiv1404.7221L"
    }
  }
}