  Assuming the Riemann Hypothesis, we combine powerful theoretical methods,
computational techniques, and the latest explicit knowledge on the zeros of the
Riemann zeta-function to prove $$|\psi(x) - x| \leq
\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$$ for all $x\geq
401\,211\,790$, where $\psi(x)$ is the Chebyshev $\psi$-function. These bounds
for the error term in the prime number theorem refine classical results from
Schoenfeld. We also demonstrate how our results can be used to refine bounds
for other prime counting functions, which have broad applications throughout
the literature.
