Explicit Bound of $\pmb{|\zeta\left(1+it\right)|}$

Eunice Hoo Qingyi; Lee-Peng Teo

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arXiv · 2412.00766

Explicit Bound of $\pmb{|\zeta\left(1+it\right)|}$

Eunice Hoo Qingyi·Lee-Peng Teo

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Abstract

In this work, we show that for all $t\geq e$, \[|\zeta(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|\zeta(1+it)|\leq\frac{1}{2}\log t+0.6633\hspace{1cm}\text{when}\;t\geq e.\]When $t\geq 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$.

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Eunice Hoo Qingyi, Lee-Peng Teo. 2024-12-01. Explicit Bound of $\pmb{|\zeta\left(1+it\right)|}$. https://arxiv.org/abs/2412.00766

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