Explicit estimates for the Riemann zeta function close to the $1$-line

Abstract

We provide explicit upper bounds of the order $\log t/\log \log t$ for $|{\zeta }^{\prime }(s)/\zeta (s)|$ and $|1/\zeta (s)|$ when $\sigma$ is close to $1$. These improve existing bounds for $\zeta (s)$ on the $1$-line.