{
  "id": "2208.08421",
  "version": "v1",
  "published": "2022-08-17T17:35:34.000Z",
  "updated": "2022-08-17T17:35:34.000Z",
  "title": "A weighted one-level density of the non-trivial zeros of the Riemann zeta-function",
  "authors": [
    "Sandro Bettin",
    "Alessandro Fazzari"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\\zeta(\\frac12+it)|^{2k}$ for $k=1$ and, for test functions with Fourier support in $(-\\frac12,\\frac12)$, for $k=2$. As a consequence, for $k=1,2$, we deduce under the Riemann hypothesis that $T(\\log T)^{1-k^2+o(1)}$ non-trivial zeros of $\\zeta$, of imaginary parts up to $T$, are such that $\\zeta$ attains a value of size $(\\log T)^{k+o(1)}$ at a point which is within $O(1/\\log T)$ from the zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-17T17:35:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26"
    ],
    "keywords": [
      "non-trivial zeros",
      "weighted one-level density",
      "riemann zeta-function",
      "test functions",
      "fourier support"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}