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

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, Valås, and Simonič.