Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function

Abstract

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. We correct an error in the literature, and especially in the case of $|1/\zeta(\sigma +it)|$, also provide improvements in the constants. Using an argument involving the trigonometric polynomial, we additionally provide a slight asymptotic improvement within the classical zero-free region: $1/\zeta(\sigma +it) \ll (\log t)^{11/12}$. The same method applied to the Korobov--Vinogradov zero-free region gives a new record: the unconditional bound $1/\zeta(\sigma +it) \ll (\log t)^{2/3}(\log\log t)^{1/4}$.