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

TL;DR

This work develops explicit conditional bounds for the Riemann zeta-function and its logarithmic derivative on the line $\Re s=1$ under the Riemann hypothesis. By marrying the Guinand–Weil explicit formula with Beurling–Selberg bandlimited extremal approximations to the Poisson kernel, the authors obtain refined lower-order terms in the classical Littlewood-type bounds for $|\zeta(1+it)|$, $|1/\zeta(1+it)|$, and $|\zeta'(1+it)/\zeta(1+it)|$, with explicit constants. They also provide sharp upper bounds for $|\zeta(1+it)|$ and a strong bound for $|\zeta'/\zeta(1+it)|$ incorporating a reduced- order term in $\log\log t$, all valid for $t\ge e^{18}$. The results sharpen the quantitative understanding of the edge of the critical strip, and improve prior conditional bounds via explicit extremal constructions and optimized parameter choices. The methods have potential applications to explicit bounds in related families of $L$-functions and to refined zero-density-type analyses in analytic number theory.

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, Valås, and Simonič.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof