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

Abstract

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 $|\psi(x) - x|$ and $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$$ for all $x\geq 101$ and $x \geq 2\,657$ 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.