On density of the zeros of Dedekind zeta-functions

TL;DR

This paper provides new bounds on the density of zeros of Dedekind zeta-functions within certain regions, improving previous results for specific ranges of the real part of zeros.

Contribution

It establishes sharper upper bounds for the number of zeros of Dedekind zeta-functions for k ≥ 3, refining earlier estimates in specified regions.

Findings

Derived bounds for zero density that depend on the parameter k and the real part of zeros.

Improved previous zero density estimates for Dedekind zeta-functions when k ≥ 3.

Bounds are valid for σ in the range [(2k+3)/(2k+6), 1).

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 $\sigma$.