An explicit log-free zero density estimate for the Riemann zeta-function

Abstract

We will provide an explicit log-free zero-density estimate for $\hat{I}\P (s)$ of the form $N(\ddot{I}ƒ,T)\le A{T}^{B(1-\ddot{I}ƒ)}$. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for $\ddot{I}ƒ\in [\hat{I}{\pm }_0,1]$, with $0.985\le \hat{I}{\pm }_0\le 0.9927$ and $3\cdot 10^{12}<T\le \exp (6.7\cdot 10^{12})$.