Sharper bounds for the Chebyshev function $\psi(x)$

Abstract

We improve the unconditional explicit bounds for the error term in the prime counting function $\psi(x)$. In particular, we prove that, for all $x>2$, we have \[ \left| \psi(x)-x \right| < 9.22106 \, x \, (\log x)^{3/2} \exp(-0.8476836\sqrt{\log x}), \] and that, for all $\log x \ge 3\,000$, \[ \left| \psi(x)-x \right| < 4.47\cdot 10^{-15} x. \] This compares to results of Platt \& Trudgian (2021) who obtained $4.51\cdot 10^{-13} x $. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning $\pi(x)$ will appear in a follow up work.