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

TL;DR

This paper provides sharper explicit bounds for the error term in the prime counting function (x), improving previous results through refined zero analysis and computational techniques, with implications for prime distribution estimates.

Contribution

The paper introduces significantly improved unconditional bounds for (x) error term, refining previous methods by splitting zeros into regions and employing advanced computational estimates.

Findings

New explicit bounds for (x) error term for all x>2.

Improved bounds for (x) when 1000.

Comparison showing tighter bounds than previous work by Platt & Trudgian.

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.