{
  "id": "2211.14625",
  "version": "v1",
  "published": "2022-11-26T17:48:33.000Z",
  "updated": "2022-11-26T17:48:33.000Z",
  "title": "On the logarithmic derivative of characteristic polynomials for random unitary matrices",
  "authors": [
    "Fan Ge"
  ],
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $U\\in U(N)$ be a random unitary matrix of size $N$, distributed with respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic polynomial of $U$. We prove that for $z$ close to the unit circle, $ \\frac{P'}{P}(z) $ can be approximated using zeros of $P$ very close to $z$, with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta-function. We also prove a mesoscopic central limit theorem for $ \\frac{P'}{P}(z) $ away from the unit circle, and this is an analogue of a result of Lester for zeta.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-26T17:48:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random unitary matrix",
      "characteristic polynomial",
      "logarithmic derivative",
      "unit circle",
      "mesoscopic central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}