{
  "id": "2102.04663",
  "version": "v1",
  "published": "2021-02-09T06:12:24.000Z",
  "updated": "2021-02-09T06:12:24.000Z",
  "title": "Counting Zeros of Dedekind Zeta Functions",
  "authors": [
    "Elchin Hasanalizade",
    "Quanli Shen",
    "Peng-Jie Wong"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros, with height at most $T$, of the Dedekind zeta function $\\zeta_K(s)$ of $K$. More precisely, we show that $$ \\Big| N_K (T) - \\frac{T}{\\pi} \\log \\Big( d_K \\Big( \\frac{T}{2\\pi e}\\Big)^{n_K}\\Big)\\Big| \\le 0.228 (\\log d_K + n_K \\log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri-Ng and Trudgian. The improvement is based on ideas from the recent work of Bennett \\emph{et al.} on counting zeros of Dirichlet $L$-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-09T06:12:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dedekind zeta function",
      "counting zeros",
      "explicit bound",
      "non-trivial zeros",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}