On density of the zeros of Dedekind zeta-functions

Abstract

For any \(\sigma \) with \(0\le \sigma \le 1\) 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 \(|\gamma |\le T\) and \(\beta \ge \sigma \) and the zero being counted according  to multiplicity. For \(k\ge 3\), we have

$$ N_{\zeta }(\sigma ,K,T)\ll T^{\frac{2k}{6\sigma -3}(1-\sigma )+\varepsilon }, $$

where

$$ \frac{2k+3}{2k+6}\le \sigma <1 $$

and the implied constant may depend on the number field K and \(\varepsilon \). This improves previous results for \(k\ge 3\) of certain range of \(\sigma \).