Arithmetic of Catalan's constant and its relatives

TL;DR

This paper proves that among six specific values of the Dirichlet beta function, at least one is irrational, advancing understanding of the arithmetic nature of Catalan's constant and related constants.

Contribution

It establishes the irrationality of at least one of six particular beta function values, a novel result in the study of special constants.

Findings

At least one of the six beta values is irrational.

The result applies to beta(2i) for i=1,...,6.

Advances knowledge of the arithmetic properties of Catalan's constant.

Abstract

We prove that at least one of the six numbers $\beta(2i)$ for $i=1,\dots,6$ is irrational. Here $\beta(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes Dirichlet's beta function, so that $\beta(2)$ is Catalan's constant.