Value of Apéry's Constant

Unsolved Problem
Apéry's constant:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

has not been determined in closed form in terms of other familiar constants.

It is of particular interest whether, similar to Riemann Zeta Function at Even Integers:


 * $\dfrac {\map \zeta 3} {\pi^3}$

is rational.

By using the techniques of Riemann Zeta Function as a Multiple Integral it can be established that:


 * $\displaystyle \int_0^1 \int_0^1 \int_0^1 \frac {\rd x \rd y \rd z} {1 - x y z} = \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

Also see

 * Apéry's Theorem, in which Apéry's constant is shown to be irrational.