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

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.