Explicit zero density for the Riemann zeta function

TL;DR

This paper provides explicit upper bounds for the number of nontrivial zeros of the Riemann zeta function with real part greater than a given , improving previous estimates by refining constants.

Contribution

It generalizes and improves existing methods to derive explicit zero density bounds for the Riemann zeta function with better constants.

Findings

Derived explicit bounds for zero density $N(\sigma,T)$

Improved constants in zero density estimates

Extended previous asymptotic results with explicit versions

Abstract

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(\sigma,T)=\mathcal{O} ( T^{\frac83(1-\sigma)} (\log T)^5 )$. Ramar\'{e} recently proved an explicit version of this estimate. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants.