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

TL;DR

This paper presents a new explicit log-free zero-density estimate for the Riemann zeta-function, providing the sharpest bounds in a specific range of the critical strip, with implications for understanding its zeros.

Contribution

It introduces the first explicit log-free zero-density estimate for ta(s) that is uniformly sharp across a broad range of ta(s) values.

Findings

Provides the explicit estimate N(ta,T)\u2264 AT^{B(1-ta)}

Achieves the sharpest known bounds for ta in [0.985, 0.9927]

Valid for T up to xp(6.7^{12})

Abstract

We will provide an explicit log-free zero-density estimate for $\zeta(s)$ of the form $N(\sigma,T)\le AT^{B(1-\sigma)}$. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for $\sigma\in[\alpha_0,1]$, with $0.985\le \alpha_0\le 0.9927$ and $3\cdot 10^{12}<T\le \exp(6.7\cdot 10^{12})$.