Question is pretty much the title. It is pretty easy to show that $\zeta(2n)$ is irrational for all $n$ once you know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (and then also use the fact that $\pi$ is transcendental or some other related result). My question is, does anyone know of a proof that $\zeta(2)$ is irrational without evaluating it as $\frac{\pi^2}{6}$ and using the fact that $\pi^2$ is irrational? Perhaps one that uses its series definition or integral representations?
Not homework, just curious.
Beukers proved $\zeta(2)$ irrational without using $\zeta(2)=\pi^2/6$. An exposition of his proof is here.