{
  "id": "2406.14987",
  "version": "v1",
  "published": "2024-06-21T08:59:54.000Z",
  "updated": "2024-06-21T08:59:54.000Z",
  "title": "Density theorems for Riemann's auxiliary function",
  "authors": [
    "Juan Arias de Reyna"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a density theorem for the auxiliar function $\\mathop{\\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\\alpha$ be a real number with $\\frac12< \\alpha\\le 1$, and let $N(\\alpha,T)$ be the number of zeros $\\rho=\\beta+i\\gamma$ of $\\mathop{\\mathcal R}(s)$ with $1\\ge \\beta\\ge\\alpha$ and $0<\\gamma\\le T$. Then we prove \\[N(\\alpha,T)\\ll T^{\\frac32-\\alpha}(\\log T)^3.\\] Therefore, most of the zeros of $\\mathop{\\mathcal R}(s)$ are near the critical line or to the left of that line. The imaginary line for $\\pi^{-s/2}\\Gamma(s/2)\\mathop{\\mathcal R}(s)$ passing through a zero of $\\mathop{\\mathcal R}(s)$ near the critical line frequently will cut the critical line, producing two zeros of $\\zeta(s)$ in the critical line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-21T08:59:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30D99"
    ],
    "keywords": [
      "riemanns auxiliary function",
      "density theorem",
      "critical line",
      "auxiliar function",
      "imaginary line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}