Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis (2312.05628v3)

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 $$|\psi(x) - x| \leq \frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$$ for all $x\geq 101$, where $\psi(x)$ is the Chebyshev $\psi$-function. Second, we prove explicit descriptions for the error in each of Mertens' theorems which remove smaller order terms from earlier bounds by Schoenfeld.