Value of Apéry's Constant

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

has not been determined.

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


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

Also see

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