{
  "id": "1201.2633",
  "version": "v2",
  "published": "2012-01-12T18:10:25.000Z",
  "updated": "2015-12-01T10:08:44.000Z",
  "title": "On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function",
  "authors": [
    "A. S. Fokas",
    "J. Lenells"
  ],
  "comment": "103 pages",
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\\zeta(s)$, $s=\\sigma+i t$, $0\\leq \\sigma \\leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $\\Phi(u,v,\\beta)$, $u\\in \\mathbb{C}$, $v\\in \\mathbb{C}$, $\\beta \\in \\mathbb{R}$. Generalizing the methodology used in the study of $\\zeta(s)$, we derive asymptotic formulae for $\\Phi(u,v,\\beta)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-12T18:10:25.000Z",
      "title": "On the Asymptotics of the Riemann Zeta Function to all Orders",
      "abstract": "We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\\zeta(s)$, $s=\\sigma+i t$, $0\\leq \\sigma \\leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. Finally, we derive certain novel integral representations for the basic sum characterising the leading large $t$ asymptotics of $\\zeta(s)$.",
      "comment": "62 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-01T10:08:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30E15",
      "33E20"
    ],
    "keywords": [
      "riemann zeta function",
      "asymptotics",
      "novel integral representations",
      "results coincides",
      "basic sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 103,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}