On density of the zeros of Dedekind zeta-functions
Wei Zhang
TL;DR
This work studies the zeros of Dedekind zeta-functions $\zeta_K(s)$ for number fields $K$ of degree $k$, introducing $N_{\zeta}(\sigma,K,T)$ to count zeros with real part at least $\sigma$ and imaginary part up to $T$. It develops a zero-detecting method using a truncated Dirichlet series $M_X(s,K)$ and derived coefficients, partitions zeros into two classes, and combines Huxley-type subdivision with exponent-pair bounds to prove a density bound $N_{\zeta}(\sigma,K,T) \ll T^{\frac{2k}{6\sigma-3}(1-\sigma)+\varepsilon}$ for $(2k+3)/(2k+6) \le \sigma < 1$ (valid for $k\ge 3$). The results improve upon Heath-Brown and other prior bounds in certain ranges of $\sigma$, broadening understanding of zero distribution for Dedekind zeta-functions. The methods have potential implications for subconvexity-type questions and the arithmetic of number fields by sharpening how zeros are distributed in the critical strip.
Abstract
For any $σ$ with $0\leq σ\leq 1$ and any $T>10$ sufficiently large, let $N_ζ(σ,K,T)$ be the number of zeros $ρ=β+iγ$ of $ζ_{K}(s)$ with $|γ|\leq T$ and $β\geq σ$ and the zero being counted according to multiplicity. For $k\geq3,$ 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 $\varepsilon.$ This improves previous results for $k\geq3$ of certain range of $σ$.