Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function
Nicol Leong
TL;DR
This work provides explicit, near-line bounds for the logarithmic derivative $\left|\frac{\zeta'(s)}{\zeta(s)}\right|$ and the reciprocal $\left|\frac{1}{\zeta(s)}\right|$ in regions close to the critical line, including corrections to prior results and improved constants. By combining zero-free-region techniques with a non-negative trigonometric polynomial and Borel–Carathéodory type bounds, the authors obtain tight $O(\log t)$ bounds and, in the classical zero-free region, an asymptotic improvement to $(\log t)^{11/12}$ for $|1/\zeta(s)|$, while in the Korobov–Vinogradov region they achieve the unconditional bound $|1/\zeta(\sigma+it)| \ll (\log t)^{2/3}(\log\log t)^{1/4}$. The paper also provides RH-dependent refinements, allowing removal of certain $t$-restrictions, and presents a detailed methodological framework including preliminary lemmas (notably Borel–Carathéodory) and an asymptotically sharper route for the reciprocal bound. These results have practical implications for explicit analytic computations in prime number theory, Perron formulas, and estimates of arithmetic functions near the 1-line. Overall, the work advances explicit zeta-analytic bounds with improved constants and new asymptotic regimes.
Abstract
In this article, we give explicit bounds of order $\log t$ for $σ$ close to $1$, for two quantities: $|ζ'(σ+it)/ζ(σ+it)|$ and $|1/ζ(σ+it)|$. We correct an error in the literature, and especially in the case of $|1/ζ(σ+it)|$, 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: $1/ζ(σ+it) \ll (\log t)^{11/12}$. The same method applied to the Korobov--Vinogradov zero-free region gives a new record: the unconditional bound $1/ζ(σ+it) \ll (\log t)^{2/3}(\log\log t)^{1/4}$.