Sharper bounds for the error term in the Prime Number Theorem

TL;DR

This paper develops effective methods to improve bounds on the error term in the Prime Number Theorem, providing sharper estimates for prime counting functions and extending numerical bounds to very large values of x.

Contribution

It introduces new techniques to convert bounds on $\psi(x)$ to bounds on $ heta(x)$ and $\pi(x)$, achieving the best numerical bounds up to extremely large x.

Findings

Proves a new explicit bound on |π(x) - Li(x)| involving x and log(x).

Extends numerical bounds for prime counting functions up to exp(1.8×10^9).

Provides effective methods for converting bounds between prime counting functions.

Abstract

We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $\psi(x)$ to bounds of the same type on both $\theta(x)$ and $\pi(x)$. This follows up our previous work on $\psi(x)$ in \cite{FKS}, and prove that $ | \pi(x) - \mathrm{Li}(x) | \leq 9.2211\, x\sqrt{\log(x)} \exp \big( -0.8476 \sqrt{\log(x)} \big) $ for all $x\ge 2$. Additionally, we are able to obtain the best numeric bounds for $x$ on a very large interval (all $x$ up to $\exp(1.8\cdot10^9)$).