Explicit conditional bounds for $\zeta(s)$ at the edge of the critical strip

TL;DR

This paper derives explicit bounds for the Riemann zeta-function's logarithmic derivative on the line Re s=1 under the Riemann hypothesis, refining classical estimates and improving lower-order terms using advanced analytical techniques.

Contribution

It introduces explicit bounds for the zeta function's logarithmic derivative on the critical line, combining the explicit formula with extremal functions, and refines existing classical estimates.

Findings

Explicit bounds for the logarithmic derivative of ζ(s) at Re s=1.

Refined estimates of ζ(s) modulus on the line Re s=1.

Improved lower-order terms in bounds under the Riemann hypothesis.

Abstract

In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line $\re s=1$, assuming the Riemann hypothesis. The proof combines the Guinand--Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line $\re{s}=1$, and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line $\re{s}=1$ under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Val{\aa}s, and Simoni\v{c}.