Explicit Bound of $|\hat{I}\P \left(1+it\right)|$

Abstract

In this work, we show that for all $t\ge e$, \[|ζ(1+it)|\leq 0.6443 \log t. \] 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 $t\ge 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$.

28 pages, 5 figures, 3 tables