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 $\sigma$.
Categories: math.NT
