{
  "id": "2406.13278",
  "version": "v1",
  "published": "2024-06-19T07:14:50.000Z",
  "updated": "2024-06-19T07:14:50.000Z",
  "title": "Mean Values of the auxiliary function",
  "authors": [
    "Juan Arias de Reyna"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathop{\\mathcal R}(s)$ be the function related to $\\zeta(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \\[\\frac{1}{T}\\int_0^T |\\mathop{\\mathcal R}(\\sigma+it)|^2\\Bigl(\\frac{t}{2\\pi}\\Bigr)^\\sigma\\,dt, \\quad\\text{and}\\quad \\frac{1}{T}\\int_0^T |\\mathop{\\mathcal R}(\\sigma+it)|^2\\,dt.\\] Giving complete proofs of some result of the paper of Siegel about the Riemann Nachlass. Siegel follows Riemann to obtain these mean values. We have followed a more standard path, and explain the difficulties we encountered in understanding Siegel's reasoning.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T07:14:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "30D99"
    ],
    "keywords": [
      "mean values",
      "auxiliary function",
      "standard path",
      "main terms",
      "riemann nachlass"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}