Value of Apéry's Constant

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Unsolved Problem

Apéry's constant:

$\displaystyle \map \zeta 3 = \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

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:

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


Historical Note

The value of Apéry's constant remains unsolved since Leonhard Paul Euler raised the question in $1736$.


Also see


Sources