On density of the zeros of Dedekind zeta-functions

Abstract: For any $\sigma$ with $0\leq \sigma\leq 1$ and any $T>10$ sufficiently large, let $N_{\zeta}(\sigma,K,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\zeta_{K}(s)$ with $|\gamma|\leq T$ and $\beta\geq \sigma$ and the zero being counted according to multiplicity. For $k\geq3,$ we have [ N_{\zeta}(\sigma,K,T)\ll T^{\frac{2k}{6\sigma-3}(1-\sigma)+\varepsilon}, ] where [ \frac{2k+3}{2k+6}\leq \sigma<1 ] and the implied constant may depend on the number field $K$ and $\varepsilon.$ This improves previous results for $k\geq3$ of certain range of $\sigma$.