  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
$\psi(x)$.
