Sharper bounds for the error term in the Prime Number Theorem

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