A new upper bound for $|\zeta(1+ it)|$

TL;DR

This paper establishes a new explicit upper bound for the magnitude of the Riemann zeta function on the line 1+it, improving previous estimates and providing the best bound for very large t up to 10^{2 o 10^{5}}.

Contribution

It introduces a novel explicit upper bound for |ig(1+it)|, surpassing previous asymptotic estimates for large t.

Findings

New explicit bound: |ig(1+it)|  3/4 \, ext{log} t

Best known upper bound for t  10^{2 o 10^{5}}

Improves understanding of zeta function growth near 1

Abstract

It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+ it)|$ for $t\leq 10^{2\cdot 10^{5}}$.