Beukers-like proofs of irrationality for $\zeta{(2)}$ and $\zeta{(3)}$

TL;DR

This paper presents modified Beukers-like proofs of the irrationality of zeta(2) and zeta(3), aiming to aid understanding and potential extensions to other constants.

Contribution

It introduces specific modifications to existing Beukers-based proofs, enhancing clarity and facilitating extensions to higher zeta values and related constants.

Findings

Proofs confirm irrationality of zeta(2) and zeta(3).

Modifications improve understanding of Beukers-like proofs.

Potential for extending methods to other mathematical constants.

Abstract

In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications which certainly will be useful for those interested in understanding this kind of proof and/or trying to extend it to higher zeta values, Catalan's constant, or other related numbers.