Counting zeros of Dedekind zeta functions

Abstract

Given a number field K K of degree n K n_K and with absolute discriminant d K d_K , we obtain an explicit bound for the number N K ( T ) N_K(T) of non-trivial zeros (counted with multiplicity), with height at most T T , of the Dedekind zeta function ζ K ( s ) \zeta _K(s) of K K . More precisely, we show that for T ≥ 1 T \geq 1 , | N K ( T ) − T π log ⁡ ( d K ( T 2 π e ) n K ) | ≤ 0.228 ( log ⁡ d K + n K log ⁡ T ) + 23.108 n K + 4.520 , \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 L -functions.

Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

“…Trudgian [31], and more recently Hasanalizade et al [15], provided explicit constants for C i , but these are not necessary for our purposes. It follows that…”