Counting zeros of Dedekind zeta functions

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 (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $\zeta _K(s)$ of $K$. More precisely, we show that for $T \geq 1$, \begin{equation*} \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, \end{equation*} which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet $L$-functions.