{
  "id": "2407.01028",
  "version": "v1",
  "published": "2024-07-01T07:24:05.000Z",
  "updated": "2024-07-01T07:24:05.000Z",
  "title": "An Integral representation of $\\mathop{\\mathcal R}(s)$ due to Gabcke",
  "authors": [
    "Juan Arias de Reyna"
  ],
  "comment": "5 pages 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Gabcke proved a new integral expression for the auxiliary Riemann function \\[\\mathop{\\mathcal R}(s)=2^{s/2}\\pi^{s/2}e^{\\pi i(s-1)/4}\\int_{-\\frac12\\searrow\\frac12} \\frac{e^{-\\pi i u^2/2+\\pi i u}}{2i\\cos\\pi u}U(s-\\tfrac12,\\sqrt{2\\pi}e^{\\pi i/4}u)\\,du,\\] where $U(\\nu,z)$ is the usual parabolic cylinder function. We give a new, shorter proof, which avoids the use of the Mordell integral. And we write it in the form \\begin{equation}\\mathop{\\mathcal R}(s)=-2^s \\pi^{s/2}e^{\\pi i s/4}\\int_{-\\infty}^\\infty \\frac{e^{-\\pi x^2}H_{-s}(x\\sqrt{\\pi})}{1+e^{-2\\pi\\omega x}}\\,dx.\\end{equation} where $H_\\nu(z)$ is the generalized Hermite polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-01T07:24:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30D99"
    ],
    "keywords": [
      "integral representation",
      "usual parabolic cylinder function",
      "auxiliary riemann function",
      "mordell integral",
      "shorter proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}