Counting Zeros of Dedekind Zeta Functions

Elchin Hasanalizade, Quanli Shen, Peng-Jie Wong

Published 2021-02-09Version 1

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.