Explicit zero density estimate for the Riemann zeta-function near the critical line

TL;DR

This paper provides explicit bounds for the distribution of zeros of the Riemann zeta-function near the critical line, including an explicit version of Selberg's zero density estimate and bounds for the zeta-function's moments.

Contribution

It offers the first explicit version of Selberg's zero density estimate and explicit bounds for the second moment of the zeta-function on the critical line.

Findings

Explicit zero density estimate near the critical line.

Explicit bounds for the second moment of the zeta-function.

Derived an explicit approximate functional equation.

Abstract

In 1946, A. Selberg proved $N(\sigma,T) \ll T^{1-\frac{1}{4} \left(\sigma-\frac{1}{2}\right)} \log{T}$ where $N(\sigma,T)$ is the number of nontrivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}>\sigma$ and $0<\Im\{\rho\}\leq T$. We provide an explicit version of this estimate, together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta-function on the critical line.