Irrationality of the Zeta Constants

TL;DR

This paper introduces a new technique to prove the irrationality of odd zeta constants, extending known results and providing evidence that these constants are irrational.

Contribution

A novel method for establishing the irrationality of odd zeta constants based on the known irrationality of beta constants.

Findings

Proves irrationality of certain odd zeta constants

Extends known irrationality results to new zeta values

Supports the conjecture that all odd zeta constants are irrational

Abstract

A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the zeta constants $\zeta(2n)$, where $n\geq 1$, and $\zeta(3)$ are well known, but the results on the irrationality for the zeta constants $\zeta(2n+1)$, where $n \geq 2$, are new, and these results seem to confirm that these constants are irrational numbers.