Zero-density estimates and the optimality of the error term in the prime number theorem

TL;DR

This paper links zero-free regions of the Riemann zeta function to improved error estimates in the prime number theorem, achieving near-optimal bounds under certain zero-density hypotheses.

Contribution

It refines the error term in the prime number theorem by connecting zero-free regions with zero-density estimates, improving previous bounds.

Findings

Derived an essentially optimal error term under specific zero-free region conditions.

Improved upon Pintz's previous error bounds for the prime number theorem.

Established a new relation between zero-free regions and error estimates in prime counting.

Abstract

We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term for some choices of the zero-free region. As an example, we show that if there are no zeros $\rho=\beta+it$ of $\zeta(s)$ with \begin{equation*} 1-\beta<\frac{1}{c(\log t)^{2/3}(\log\log t)^{1/3}}=:\eta(t), \end{equation*} then \begin{equation*} \frac{|\psi(x)-x|}{x}\ll\exp(-\omega(x))\frac{(\log x)^9}{(\log\log x)^3}, \end{equation*} where $\psi(x)$ is the Chebyshev prime-counting function, and \begin{equation*} \omega(x)=\min_{t\geq 3}\{\eta(t)\log x+\log t\}. \end{equation*} This refines the best known error term for the prime number theorem, previously given by \begin{equation*} \frac{|\psi(x)-x|}{x}\ll_{\varepsilon}\exp(-(1-\varepsilon)\omega(x)) \end{equation*} for any $\varepsilon>0$.