A note on Apery's constant is transcendental

Abstract

Beuker's [2] considers the following integral $${\int }_0^1{\int }_0^1\frac{-\log xy}{1-xy}{P}_n(x)P_n(y)dxdy$$If $d_n=\text{LCM}(1,2,...,n)$, then $$0<\frac{|A_n+B_n\zeta (3)|}{d_n^3}<2(\sqrt{2}-1{)}^{4n}\zeta (3)$$ for some $A_n,B_n\in \mathbb{Z}$. We establish that if Apery's constant is algebraic then the above inequality fails to be true. This proves that $\zeta (3)$ is

5 pages