{
  "id": "2406.04714",
  "version": "v1",
  "published": "2024-06-07T07:46:26.000Z",
  "updated": "2024-06-07T07:46:26.000Z",
  "title": "Asymptotic Expansions of the auxiliary function",
  "authors": [
    "Juan Arias de Reyna"
  ],
  "comment": "28 pages, 3 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Siegel in 1932 published a paper on Riemann's posthumous writings, including a study of the Riemann-Siegel formula. In this paper we explicitly give the asymptotic developments of $\\mathop{\\mathcal R }(s)$ suggested by Siegel. We extend the range of validity of these asymptotic developments. As a consequence we specify a region in which the function $\\mathop{\\mathcal R }(s)$ has no zeros. We also give complete proofs of some of Siegel's assertions. We also include a theorem on the asymptotic behaviour of $\\mathop{\\mathcal R }(\\frac12-it)$ for $t \\to+\\infty$. Although the real part of $e^{-i\\vartheta(t)}\\mathop{\\mathcal R }(\\frac12-it)$ is $Z(t)$ the imaginary part grows exponentially, this is why for the study of the zeros of $Z(t)$ it is preferable to consider $\\mathop{\\mathcal R }(\\frac12+it)$ for $t>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-07T07:46:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30D99"
    ],
    "keywords": [
      "auxiliary function",
      "asymptotic expansions",
      "asymptotic developments",
      "imaginary part grows",
      "riemann-siegel formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}