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

TL;DR

This paper establishes explicit bounds for the magnitude of the Riemann zeta function on the line ext{Re}(s)=1, providing precise inequalities and numerical verification for large t values.

Contribution

The work derives explicit upper bounds for |z(1+it)|, including a sharp bound with equality at a specific t, and compares different bounds for large t using numerical methods.

Findings

|z(1+it)|  0.6443 \,\log t for all t  e.

The bound |z(1+it)|  rac{1}{2}\,\log t + 0.6633 for t  e.

For t  100, the second bound is tighter than the first.

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 $0.6443\log t$.