COUNTING ZEROS OF DEDEKIND ZETA FUNCTIONS

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 >= 1, vertical bar N-K(T) - T/pi log (d(K) (T/2 pi e)(nK))vertical bar <= 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.

Automated summary

This paper provides an explicit bound for the number of zeros of the Dedekind zeta function of a number field K and improves previous results. The improvement is based on recent work on counting zeros of Dirichlet L-functions.