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 \(\psi(x)\).