Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function

Abstract

The purpose of this article is consider $|\zeta'(\sigma + it)/\zeta(\sigma + it)|$ and $|\zeta(\sigma + it)|^{-1}$ when $\sigma$ is close to unity. We prove that $|\zeta'(\sigma + it)/\zeta(\sigma + it)| \leq 87\log t$ and $|\zeta(\sigma + it)|^{-1} \leq 6.9\times 10^{6} \log t$ for $\sigma \geq 1-1/(8 \log t)$ and $t\geq 45$.