Apery proved in 1976 that $\zeta(3)$ is irrational, and we know that for all integers $n$, $\zeta(2n)=\alpha \pi^{2n}$ for some $\alpha\in \mathbb{Q}$. Given these facts, it seems natural to ask whether we can have $\zeta(n)=\alpha \pi^n$ for all $n$ (I'm mainly interested in the case $n=3$). The proofs I've seen of the irrationality of $\zeta(3)$ don't seem to give this information.

My gut feeling is that the answer is no, but I can't find any reference proving this fact. I know that the answer hasn't been proven to be yes ($\zeta(3)$ isn't even known to be transcendental), but ruling out this possibility seems an easier problem.

notnatural to ask whether $\zeta(3)$ is a rational multiple of $\pi^3$! There are "natural" conjectures to make about special values of a huge class of zeta functions, including Riemann's, and once you have figured out what's going on then you realise that the (conjectural) story for Riemanns zeta should be quite different at odd and even integers. Evidence for this: look at the values of Riemann's zeta atnegativeintegers! it always vanishes at $-2,-4,-6,\ldots$ and never vanishes at $-1,-3,-5,\ldots$. $\endgroup$4more comments