On the complex magnitude of Dirichlet beta function

TL;DR

This paper derives new formulas for the complex magnitude of the Dirichlet beta function, compares it with the Riemann zeta function, and explores its properties and asymptotic behavior, including connections to fundamental constants.

Contribution

It introduces novel expressions for the Dirichlet beta function's magnitude, including formulas valid at specific integers and in the critical strip, and links to constants like Euler-Mascheroni.

Findings

Derived formulas for $eta(s)$ at even and odd integers

Expressed the Euler-Mascheroni constant in terms of zeros of $eta(s)$

Analyzed the asymptotic behavior of the prime product on the critical line

Abstract

In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $\beta(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $\beta(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $\beta(s)$ valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.