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 $x\ge 101$ 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.

Big update over previous versions of the paper, wherein we have streamlined all of our arguments to improve readability of the paper