On density of the zeros of Dedekind zeta-functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866918452667613184 |
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| author | Zhang, Wei |
| author_facet | Zhang, Wei |
| contents | 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 $σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_08384 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On density of the zeros of Dedekind zeta-functions Zhang, Wei Number Theory 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 $σ$. |
| title | On density of the zeros of Dedekind zeta-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2203.08384 |