{
  "id": "2311.13554",
  "version": "v1",
  "published": "2023-11-22T17:55:23.000Z",
  "updated": "2023-11-22T17:55:23.000Z",
  "title": "A discrete mean value of the Riemann zeta function",
  "authors": [
    "Kübra Benli",
    "Ertan Elma",
    "Nathan Ng"
  ],
  "comment": "42 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this work, we estimate the sum \\begin{align*} \\sum_{0 < \\Im(\\rho) \\leq T} \\zeta(\\rho+\\alpha)X(\\rho) Y(1\\!-\\! \\rho) \\end{align*} over the nontirival zeros $\\rho$ of the Riemann zeta funtion where $\\alpha$ is a complex number with $\\alpha\\ll 1/\\log T$ and $X(\\cdot)$ and $Y(\\cdot)$ are some Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where $\\zeta(\\rho+\\alpha)$ is replaced by $\\zeta^{(m)}(\\rho)$ for all $m\\in\\mathbb{N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann Hypothesis we obtain the lower bound \\begin{align*} \\sum_{0 < \\Im(\\rho) < T} | \\zeta^{(m)}(\\rho)|^{2k} \\gg T(\\log T)^{k^2+2km+1} \\quad \\quad (k,m\\in\\mathbb{N}) \\end{align*} which was previously known under the Generalized Riemann Hypothesis, in the case $m=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-22T17:55:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11N37"
    ],
    "keywords": [
      "discrete mean value",
      "riemann zeta function",
      "riemann zeta funtion",
      "generalized riemann hypothesis",
      "complex number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}