It is well known that the values of the Riemann zeta function for even positive numbers are of the form:
$$\zeta(2k) = \rm rational * \pi ^{2k},$$
and more specifically $\zeta (2k)=(-1)^{{k+1}}{\frac {B_{{2k}}(2\pi )^{{2k}}}{2(2k)!}}\!$. It is not that far-fetched to consider that
$$\zeta(2k + 1) = \rm rational * \pi ^{2k + 1}.$$
Specifically for Apéry's constant (which is $\zeta(3)$), did someone prove something like that? The proof should be something like:
$\frac{\zeta(3)}{\pi^3}$ is rational / irrational / transcendental.
EDIT: Even if the question is still open (which I can see it is from the comments), is there any new development on this matter lately? Just curious.