Counting zeros of Dedekind zeta functions

Abstract

Given a number field K of degree nK and with absolute discriminant dK , we obtain an explicit bound for the number NK (T ) of non-trivial zeros, with height at most T , of the Dedekind zeta function ζK (s) of K. More precisely, we show that NK (T ) − T π log dK T 2πe n K ≤ 0.228(log dK + nK log T ) + 23.108nK + 4.520, which improves previous results of Kadiri-Ng and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet L-functions.

Explicit Interval Estimates for Prime Numbers

“…There is no generalised Riemann height, so we can only use zero-free and zero-density regions of ζ K to obtain this estimate. At the time of writing, the second author provides the latest zero-free results in [23], and Trudgian provides the latest peer-reviewed zero-density results in [39], soon to be superseded by Hasanalizade et al [17].…”