Bounding $\zeta(s)$ in the critical strip

TL;DR

Under the assumption of the Riemann Hypothesis, the paper derives explicit bounds for the logarithm of the Riemann zeta function within the critical strip using extremal functions for a specific logarithmic expression.

Contribution

It introduces new extremal majorants and minorants for a logarithmic function to establish explicit bounds for ext{log}| ext{ extit{ extzeta}}( ext{ extit{ extalpha}} + it)| in the critical strip, extending prior work.

Findings

Derived explicit bounds for ext{log}| ext{ extit{ extzeta}}( ext{ extit{ extalpha}} + it)| assuming RH.

Extended previous bounds by Chandee and Soundararajan using new extremal functions.

Provided explicit constants for the bounds in the critical strip.

Abstract

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log\left(\tfrac{4 + x^2}{(\alpha-1/2)^2 + x^2}\right)$ to find upper and lower bounds with explicit constants for $\log|\zeta(\alpha + it)|$ in the critical strip, extending the work of Chandee and Soundararajan \cite{CS}.