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

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $\mathrm{\mid }{\zeta }^{\mathrm{\prime }}(\sigma +it)\mathrm{/}\zeta (\sigma +it)\mathrm{\mid }$ and $\mathrm{\mid }1\mathrm{/}\zeta (\sigma +it)\mathrm{\mid }$. We correct an error in the literature, and especially in the case of $\mathrm{\mid }1\mathrm{/}\zeta (\sigma +it)\mathrm{\mid }$, 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: $1\mathrm{/}\zeta (\sigma +it)\ll (\log t{)}^{11\mathrm{/}12}$. The same method applied to the Korobov–Vinogradov zero-free region gives a new record: the unconditional bound $1\mathrm{/}\zeta (\sigma +it)\ll (\log t{)}^{2\mathrm{/}3}(\log \log t{)}^{1\mathrm{/}4}$.