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

TL;DR

This paper leverages recent verification of the Riemann hypothesis up to a large height to derive improved bounds on prime counting functions for extensive ranges of x, refining previous estimates and applying to Ramanujan's inequality.

Contribution

It provides new bounds on prime counting functions based on partial verification of the Riemann hypothesis, extending the range of x where these bounds hold.

Findings

Established that |rithm(x)-li(x)|<rithm(x) for 2657  x  1.101 10^{26}

Derived bounds on rithm(x) for wider x ranges

Applied bounds to Ramanujan's inequality

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 $2657\leq x\leq 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.