Note on the $a$-points of the Riemann zeta function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929599510740992 |
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| author | Hang, Peng-Cheng Luo, Min-Jie |
| author_facet | Hang, Peng-Cheng Luo, Min-Jie |
| contents | For any $a\in\mathbb{C}$, the zeros of $ζ(s)-a$, denoted by $ρ_a=β_a+iγ_a$, are called $a$-points of the Riemann zeta function $ζ(s)$. In this paper, we reformulate some basic results about the $a$-points of $ζ(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,δ)=\sum_{τ<γ_a\leqslant T}ζ'(ρ_a+iδ)X^{ρ_a},\quad T\to\infty,\] where $0\neδ=\frac{2πα}{\log\frac{T}{2πX}}\ll 1$, and $X>0$ and $τ\geqslant|δ|+1$ are fixed.
We also find the interesting varied behavior of $S_T(a,δ)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Note on the $a$-points of the Riemann zeta function Hang, Peng-Cheng Luo, Min-Jie Number Theory 11M06, 11M26, 41A60 For any $a\in\mathbb{C}$, the zeros of $ζ(s)-a$, denoted by $ρ_a=β_a+iγ_a$, are called $a$-points of the Riemann zeta function $ζ(s)$. In this paper, we reformulate some basic results about the $a$-points of $ζ(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,δ)=\sum_{τ<γ_a\leqslant T}ζ'(ρ_a+iδ)X^{ρ_a},\quad T\to\infty,\] where $0\neδ=\frac{2πα}{\log\frac{T}{2πX}}\ll 1$, and $X>0$ and $τ\geqslant|δ|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,δ)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump. |
| title | Note on the $a$-points of the Riemann zeta function |
| topic | Number Theory 11M06, 11M26, 41A60 |
| url | https://arxiv.org/abs/2411.13255 |