{
  "id": "2407.18859",
  "version": "v1",
  "published": "2024-07-26T16:47:54.000Z",
  "updated": "2024-07-26T16:47:54.000Z",
  "title": "Tauberian theory and the Riemann hypothesis",
  "authors": [
    "Benoit Cloitre"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, I present a Tauberian equivalence of the Riemann hypothesis within the framework of the theory of regular arithmetic functions, a branch of Tauberian theory that extends the theory of functions with good variation introduced in [Cloitre]. The central element of this study is the function $\\Phi(x)=x\\left\\lfloor \\frac{1}{x}\\right\\rfloor$, which allows for the extension of Ingham's summation method [Ingham] far beyond the prime number theorem by linking it to the Riemann hypothesis. I thus demonstrate the following equivalence $$RH\\Longleftrightarrow\\alpha\\left(\\Phi\\right)=\\frac{1}{2}$$ where $\\alpha\\left(\\Phi\\right)$ represents the index of good variation of $\\Phi$, an essential characteristic of functions of good variation. This equivalence of the Riemann hypothesis is a potential new contribution absent from recent comprehensive catalogs of equivalent of the Riemann hypothesis in mathematical literature [Broughan1, Broughan2].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-26T16:47:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11N37",
      "40E05",
      "11M41"
    ],
    "keywords": [
      "riemann hypothesis",
      "tauberian theory",
      "regular arithmetic functions",
      "inghams summation method",
      "prime number theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}