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

In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that $\mathrm{\mid }\psi (x)-x\mathrm{\mid }$ and $\mathrm{\mid }\vartheta (x)-x\mathrm{\mid }$ are bounded from above by $$\frac{\sqrt{x}\log x(\log x-\log \log x)}{8\pi }$$ for all $x\ge 101$ and $x\ge 2657$ respectively, where $\psi (x)$ and $\vartheta (x)$ are the Chebyshev $\psi$ and $\vartheta$ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens’ theorems which improve earlier bounds by Schoenfeld.