Will there be a closed form expression for all $\zeta(2n+1)$?

Question

It is known that $\zeta(2)=\frac{\pi^{2}}{6}$ and that $\zeta(4)=\frac{\pi^{4}}{90}$. Thus, for $\zeta(2n)$, this can be generalized to : $$ \zeta(2n)=\frac{(-1)^{n-1}B_{2n}(2\pi)^{2n}}{2(2n)!} $$ Where $B_{2n}$ are the Bernoulli's numbers. Now why is that such closed form expression does not exist for all $\zeta(2n+1)$?

Answer 1

In general, it is an open conjecture whether or not $\zeta(2n+1)$ is a rational multiple of $\pi^{2n+1}$. This not even clear for $n=1$. The following series representation was found by Ramanujan: $$ \zeta (3)=\frac {7}{180}\pi ^{3}-2\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}. $$ His formula for $\zeta(2n+1)$ can be found here.