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

TL;DR

This paper provides improved explicit bounds on the error terms in the prime number theorem using advanced techniques related to the Riemann zeta-function, enhancing accuracy for large x.

Contribution

It introduces new explicit estimates for error terms in prime number theorem, leveraging improved zero-free regions and zero-density estimates of the Riemann zeta-function.

Findings

Explicit bound for | heta(x)-x| with a specific constant.

Explicit bound for | ext{psi}(x)-x| with improved constants.

Enhanced bounds applicable for large x values.

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 $x\geq 2$ that $|\psi(x)-x|\leq 9.39x(\log x)^{1.515}\exp(-0.8274\sqrt{\log x})$. 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 $x$.