{
  "id": "2201.06184",
  "version": "v2",
  "published": "2022-01-17T02:51:59.000Z",
  "updated": "2022-03-07T11:02:14.000Z",
  "title": "On the average value of $π(t)-\\text{li}(t)$",
  "authors": [
    "Daniel R. Johnston"
  ],
  "comment": "11 pages; to appear in Canad. Math. Bull",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that the Riemann hypothesis is equivalent to the condition $\\int_{2}^x\\left(\\pi(t)-\\text{li}(t)\\right)\\mathrm{d}t<0$ for all $x>2$. Here, $\\pi(t)$ is the prime-counting function and $\\text{li}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function $\\theta(t)$ and discuss the extent to which one can make related claims unconditionally.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-07T11:02:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11N05"
    ],
    "keywords": [
      "average value",
      "chebyshev function",
      "riemann hypothesis",
      "logarithmic integral",
      "prime-counting function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}