An explicit version of Carlson's theorem

TL;DR

This paper presents an explicit and improved version of Carlson's zero density estimate for the Riemann zeta function, providing clearer bounds on the distribution of its zeros.

Contribution

It offers a more precise explicit bound for the number of zeros of the Riemann zeta function, refining previous estimates with a slight improvement in the logarithmic factor.

Findings

Explicit zero density estimate with improved logarithmic exponent

Provides bounds for zeros with real part greater than 6

Enhances understanding of the zero distribution of the zeta function

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 lying between $0$ and $T$. In this article, we provide an explicit version of Carlson's zero density estimate, that is, $N(\sigma, T) \leq 0.78 T^{4 \sigma (1- \sigma)} (\log T)^{5-2 \sigma} $, with a slight improvement in the exponent of the logarithm factor.