Improved estimates for the argument and zero-counting function of the Riemann zeta-function
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909676327665664 |
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| author | Bellotti, Chiara Wong, Peng-Jie |
| author_facet | Bellotti, Chiara Wong, Peng-Jie |
| contents | In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing
\[
\left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344,
\] for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $ρ=β+iγ$, with $0<γ\le T$, of the Riemann zeta-function $ζ(s)$. The main source of improvement comes from implementing new subconvexity bounds for $ζ(σ+it)$ on some $σ_k$-lines inside the critical strip. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved estimates for the argument and zero-counting function of the Riemann zeta-function Bellotti, Chiara Wong, Peng-Jie Number Theory 11M06, 11M26 (Primary) 11Y35 (Secondary) In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $ρ=β+iγ$, with $0<γ\le T$, of the Riemann zeta-function $ζ(s)$. The main source of improvement comes from implementing new subconvexity bounds for $ζ(σ+it)$ on some $σ_k$-lines inside the critical strip. |
| title | Improved estimates for the argument and zero-counting function of the Riemann zeta-function |
| topic | Number Theory 11M06, 11M26 (Primary) 11Y35 (Secondary) |
| url | https://arxiv.org/abs/2412.15470 |