{
  "id": "2406.07968",
  "version": "v1",
  "published": "2024-06-12T07:44:51.000Z",
  "updated": "2024-06-12T07:44:51.000Z",
  "title": "On Siegel results about the zeros of the auxiliary function of Riemann",
  "authors": [
    "Juan Arias de Reyna"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\\mathop{\\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\\mathop{\\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\\sqrt{T/2\\pi}P(\\sqrt{T/2\\pi})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\\mathop{\\mathcal R}(s)$. We precise also the connection of the position on the zeros of $\\mathop{\\mathcal R}(s)$ with the zeros of $\\zeta(s)$ in the critical line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-12T07:44:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30D99"
    ],
    "keywords": [
      "auxiliary function",
      "siegel results",
      "complete proof",
      "positive imaginary part",
      "periodic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}