Estimating $\pi (x)$ and related functions under partial RH assumptions

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi (x)$. This is done by proving the well-known explicit Schoenfeld bound on the RH to hold as long as $4.92 \sqrt{x/\log(x)} \leq T$. Similar statements are proven for the Riemann prime-counting function and the Chebyshov functions $\psi (x)$ and $\vartheta (x)$. Apart from that, we also improve some of the existing bounds of Chebyshov type for the function $\psi (x)$.