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 $T\ge 1$, $$|N_K(T)-\frac{T}{\pi }\log (d_K\left(\frac{T}{2\pi e}{)}^{n_K}\right)|\le 0.228(\log d_K+n_K\log T)+23.108n_K+4.520,$$ 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.

Accepted by Math. Comp