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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2412.00766 |
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| _version_ | 1866918155273633792 |
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| author | Qingyi, Eunice Hoo Teo, Lee-Peng |
| author_facet | Qingyi, Eunice Hoo Teo, Lee-Peng |
| contents | In this work, we show that for all $t\geq e$, \[|ζ(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|ζ(1+it)|\leq\frac{1}{2}\log t+0.6633\hspace{1cm}\text{when}\;t\geq e.\]When $t\geq 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Bound of $\pmb{|ζ\left(1+it\right)|}$ Qingyi, Eunice Hoo Teo, Lee-Peng Number Theory In this work, we show that for all $t\geq e$, \[|ζ(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|ζ(1+it)|\leq\frac{1}{2}\log t+0.6633\hspace{1cm}\text{when}\;t\geq e.\]When $t\geq 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$. |
| title | Explicit Bound of $\pmb{|ζ\left(1+it\right)|}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.00766 |