Zero-density estimates and the optimality of the error term in the prime number theorem
Daniel R. Johnston
TL;DR
This work investigates how zero-free regions and zero-density estimates for the Riemann zeta function influence the error term in the prime number theorem. By combining a general zero-free region $\eta(t)$ with a zero-density bound $N(\sigma,t)$ and introducing the decay function $\omega(x)=\min_{t}\{\eta(t)\log x+\log t\}$, the authors prove that $\Delta_i(x) \ll \exp(-\omega(x))\exp\left(2A\omega(x)(\omega(x)/\log x)^B\right)\omega(x)^C$ under natural hypotheses, for $i=1,2,3$. They derive explicit corollaries for the classical zero-free region $\eta(t)=1/(R\log t)$ and for Vinogradov–Korobov type regions, yielding essentially optimal error terms in several regimes, notably $\Delta_i(x) \ll \exp(-\omega(x))$ up to secondary factors when $\omega(x)$ grows like $\sqrt{\log x}$ or like $(\log x)^{3/5}/(\log \log x)^{1/5}$. The results clarify the precise role of zero-density exponents and extend the understanding of unconditional PNT error terms in the presence of general zero-free regions.
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 $ρ=β+it$ of $ζ(s)$ with \begin{equation*} 1-β<\frac{1}{c(\log t)^{2/3}(\log\log t)^{1/3}}=:η(t), \end{equation*} then \begin{equation*} \frac{|ψ(x)-x|}{x}\ll\exp(-ω(x))\frac{(\log x)^9}{(\log\log x)^3}, \end{equation*} where $ψ(x)$ is the Chebyshev prime-counting function, and \begin{equation*} ω(x)=\min_{t\geq 3}\{η(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{|ψ(x)-x|}{x}\ll_{\varepsilon}\exp(-(1-\varepsilon)ω(x)) \end{equation*} for any $\varepsilon>0$.