On density of the zeros of Dedekind zeta-functions

Abstract

For any $\ddot{I}ƒ$ with $0\le \ddot{I}ƒ\le 1$ and any $T>10$ sufficiently large, let $N_{\hat{I}}\P (\ddot{I}ƒ,K,T)$ be the number of zeros $\ddot{I}={\hat{I}}^2+i{\hat{I}}^3$ of $\hat{I}{\P }_K(s)$ with $|{\hat{I}}^3|\le T$ and ${\hat{I}}^2\ge \ddot{I}ƒ$ and the zero being counted according to multiplicity. For $k\ge 3,$ we have \[ N_ζ(σ,K,T)\ll T^{\frac{2k}{6σ-3}(1-σ)+\varepsilon}, \] where \[ \frac{2k+3}{2k+6}\leq σ<1 \] and the implied constant may depend on the number field $K$ and $\epsilon .$ This improves previous results for $k\ge 3$ of certain range of $\ddot{I}ƒ$.