Is Apéry's constant a rational multiple of $ \pi ^ 3$? 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.

While Apéry proved in 1978 that $\zeta(3)$ is irrational, the irrationality of $\frac{\zeta(3)}{\pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $\pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:

$$ \begin{aligned} \zeta(3)&=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty \frac{1}{n^3(e^{2\pi n}-1)},\\\ \sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} &= -\frac{4}{3}\,\zeta(3)+\frac{\pi\sqrt{3}}{2\cdot 3^2}\,\left(\zeta(2, \tfrac{1}{3})-\zeta(2,\tfrac{2}{3}) \right). \end{aligned} $$

Moreover, in this Math.SE post we have:

$$ \frac{3}{2}\,\zeta(3) = \frac{\pi^3}{24}\sqrt{2}-2\sum_{k=1}^\infty \frac{1}{k^3(e^{\pi k\sqrt{2}}-1)}-\sum_{k=1}^\infty\frac{1}{k^3(e^{2\pi k\sqrt{2}}-1)}. $$

You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.