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

Assuming the Riemann Hypothesis, we provide effective upper and lower estimates for $\mid \zeta (s)\mid$ 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.