Explicit estimates for $\zeta(s)$ in the critical strip under the Riemann Hypothesis

TL;DR

Under the Riemann Hypothesis, the paper derives explicit bounds for the zeta function near the critical line, improving understanding of its behavior and enabling precise estimates for related number-theoretic functions.

Contribution

It provides the first explicit upper and lower bounds for |z(s)| in the critical strip assuming RH, refining previous asymptotic results.

Findings

Explicit bounds for |z(s)| near the critical line under RH

Refined bounds for the Mertens function based on zeta estimates

Improved estimates for the distribution of k-free numbers

Abstract

Assuming the Riemann Hypothesis, we provide effective upper and lower estimates for $\left|\zeta(s)\right|$ right to the critical line. As an application we make explicit Titchmarsh's conditional bound for the Mertens function and Montgomery--Vaughan's conditional bound for the number of $k$-free numbers.