{
  "id": "2008.06140",
  "version": "v1",
  "published": "2020-08-14T00:12:17.000Z",
  "updated": "2020-08-14T00:12:17.000Z",
  "title": "The mean square of the error term in the prime number theorem",
  "authors": [
    "Richard P. Brent",
    "David J. Platt",
    "Timothy S. Trudgian"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that, on the Riemann hypothesis, $\\limsup_{X\\to\\infty}I(X)/X^{2} \\leq 0.8603$, where $I(X) = \\int_X^{2X} (\\psi(x)-x)^2\\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\\frac{1}{5\\,374}\\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\\rightarrow\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-14T00:12:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11N05"
    ],
    "keywords": [
      "prime number theorem",
      "error term",
      "mean square",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}