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Assuming the Riemann Hypothesis, we combine powerful theoretical methods,computational techniques, and the latest explicit knowledge on the zeros of theRiemann zeta-function to compute $x_m$ for any positive integer $m$ such that$$|\psi(x) - x| \leq \left(\frac{\log{x}}{8\pi} - m\right) \sqrt{x}\log{x}$$for all $x\geq x_m$, where $\psi(x)$ is the Chebyshev $\psi$-function. Thesebounds for the error term in the prime number theorem refine classical resultsfrom Schoenfeld. We also demonstrate how our results can be used to refinebounds for other prime counting functions, which have broad applicationsthroughout the literature.

Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis