{
  "id": "math/0509312",
  "version": "v7",
  "published": "2005-09-14T13:34:59.000Z",
  "updated": "2009-03-23T14:06:02.000Z",
  "title": "A new bound for the smallest $x$ with $π(x) > li(x)$",
  "authors": [
    "Kuok Fai Chao",
    "Roger Plymen"
  ],
  "comment": "Final version, to be published in the International Journal of Number Theory [copyright World Scientific Publishing Company][www.worldscinet.com/ijnt]",
  "categories": [
    "math.NT"
  ],
  "abstract": "We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann zeros, we prove that there exists $x$ in the interval $[exp(727.951858), exp(727.952178)]$ for which $\\pi(x)-\\li(x) > 3.2 \\times 10^{151}$. There are at least $10^{154}$ successive integers $x$ in this interval for which $\\pi(x)>\\li(x)$. This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.",
  "revisions": [
    {
      "version": "v7",
      "updated": "2009-03-23T14:06:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11Y35",
      "11M26"
    ],
    "keywords": [
      "riemann zeros",
      "lehmans theorem",
      "main theorem",
      "leading term",
      "successive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9312C"
    }
  }
}