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

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