Sharper bounds for the error term in the Prime Number Theorem

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 $|\pi (x)-\mathrm{Li}(x)|\le 9.2211x\sqrt{\log (x)}\exp (-0.8476\sqrt{\log (x)})$ 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 $\exp (1.8\cdot 10^9)$).

arXiv comment19 pages with 7 tables, see previous arxiv version for ancillary file (167 pages) which includes more detailed tables