On density of the zeros of Dedekind zeta-functions

Wei Zhang

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arXiv · 2203.08384

On density of the zeros of Dedekind zeta-functions

Wei Zhang

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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 $σ$.

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Wei Zhang. 2025-03-14. On density of the zeros of Dedekind zeta-functions. https://arxiv.org/abs/2203.08384

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