Some explicit estimates for the error term in the prime number theorem

Abstract

By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi (x)-\text{li}(x)|$, $|\theta (x)-x|$ and $|\psi (x)-x|$ appearing in the prime number theorem. For example, we show for all $x\ge 2$ that $|\psi (x)-x|\le 9.39x(\log x{)}^{1.515}\exp (-0.8274\sqrt{\log x})$. Our estimates rely heavily on explicit zero-free regions and zero-density estimates for the Riemann zeta-function, and improve on existing bounds for prime-counting functions for large values of $x$.

22 pages. Small changes in response to feedback on first version