{
  "id": "2211.08571",
  "version": "v1",
  "published": "2022-11-15T23:29:20.000Z",
  "updated": "2022-11-15T23:29:20.000Z",
  "title": "Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line",
  "authors": [
    "Fan Ge"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number $a>0$, \\begin{align*} \\lim_{a \\to 0^+}\\lim_{T \\to \\infty} \\frac{(2a)^{2K-1}}{T (\\log T)^{2K}} \\int_{T}^{2T} \\left|\\frac{\\zeta'}{\\zeta}\\left(\\frac{1}{2}+\\frac{a}{\\log T}+it\\right)\\right|^{2K} dt = \\binom{2K-2}{K-1}. \\end{align*} When $K=1$, this was essentially a result of Goldston, Gonek and Montgomery.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-15T23:29:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta-function",
      "mean values",
      "critical line",
      "logarithmic derivative",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}