Value of Apéry's Constant
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Unsolved Problem
- $\ds \map \zeta 3 = \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$
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:
- $\ds \int_0^1 \int_0^1 \int_0^1 \frac {\rd x \rd y \rd z} {1 - x y z} = \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$
Also see
- Apéry's Theorem, in which Apéry's constant is shown to be irrational.
Historical Note
The value of Apéry's constant remains unsolved since Leonhard Paul Euler raised the question in $1736$.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.24$: Euler's Formula $\ds \sum \limits_1^\infty \frac 1 {n^2} = \frac {\pi^2} 6$ by Double Integration