  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 $t\leq 10^{2\cdot
10^{5}}$.
