Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

Abstract

Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $\pi (x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|\pi (x)-\text{li}(x)|<\sqrt{x}\log x/8\pi$ for $2657\le x\le 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.

12 pages, to appear in Ramanujan J