{
  "id": "2009.05251",
  "version": "v1",
  "published": "2020-09-11T06:43:49.000Z",
  "updated": "2020-09-11T06:43:49.000Z",
  "title": "A harmonic sum over nontrivial zeros of the Riemann zeta-function",
  "authors": [
    "Richard P. Brent",
    "David J. Platt",
    "Timothy S. Trudgian"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the sum $\\sum 1/\\gamma$, where $\\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \\to\\infty$. We show that, after subtracting a smooth approximation $\\frac{1}{4\\pi} \\log^2(T/2\\pi),$ the sum tends to a limit $H \\approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-11T06:43:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11Y30"
    ],
    "keywords": [
      "nontrivial zeros",
      "riemann zeta-function",
      "harmonic sum",
      "smooth approximation",
      "sum tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}