Counting zeros of Dedekind zeta functions

Given a number field K K of degree alttext="n Subscript upper n encoding="application/x-tex">n_K and with absolute discriminant alttext="d d encoding="application/x-tex">d_K , we obtain an explicit bound for the N K Baseline left-parenthesis T right-parenthesis"> N ( T stretchy="false">) encoding="application/x-tex">N_K(T) non-trivial zeros (counted multiplicity), height at most T"> encoding="application/x-tex">T Dedekind zeta function alttext="zeta s ? s encoding="application/x-tex">\zeta _K(s) . More precisely, show that greater-than-or-equal-to 1"> ? 1 encoding="application/x-tex">T \geq 1 maxsize="1.623em" minsize="1.623em">| ?<!-- ? <mml:mfrac> ?<!-- ? </mml:mfrac> log ?<!-- ? minsize="1.623em">( 2 e minsize="1.623em">) ?<!-- ? <mml:mn>0.228 + 23.108 4.520 , encoding="application/x-tex">\begin{equation*} \Big | N_K (T) - \frac {T}{\pi } \log ( d_K {T}{2\pi e}\Big )^{n_K}\Big )\Big \le (\log + n_K T) 4.520, \end{equation*} which improves previous results Kadiri Ng, Trudgian. The improvement is based on ideas from recent work Bennett et al. counting Dirichlet L"> L encoding="application/x-tex">L -functions.