{
  "id": "1601.05324",
  "version": "v1",
  "published": "2016-01-20T16:42:58.000Z",
  "updated": "2016-01-20T16:42:58.000Z",
  "title": "On General Prime Number Theorems with Remainder",
  "authors": [
    "Gregory Debruyne",
    "Jasson Vindas"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that for Beurling generalized numbers the prime number theorem in remainder form $$\\pi(x) = \\operatorname*{Li}(x) + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all} n\\in\\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\\left(\\frac{x}{\\log^{n}x}\\right) \\quad \\mbox{for all} n \\in \\mathbb{N},$$ where $N$ and $\\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299--307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Ces\\`aro sense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-20T16:42:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N80",
      "11M45"
    ],
    "keywords": [
      "general prime number theorems",
      "remainder form",
      "acta math",
      "subject contains",
      "average version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160105324D"
    }
  }
}