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# Apéry's constant

Apéry's constant ${\displaystyle \scriptstyle \zeta (3)\,}$, named after Roger Apéry (who proved that it is irrational), is the sum of the reciprocals of the cubes of all the positive integers

${\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}=\prod _{i=1}^{\infty }{\frac {1}{1-{\frac {1}{(p_{i})^{3}}}}},\,}$[1]

where ${\displaystyle \scriptstyle \zeta (s)\,}$ is the Riemann zeta function and, in Euler's product, ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$th prime.

It is unknown whether the constant is transcendental. It is also unknown whether there is a simple formula for ${\displaystyle \scriptstyle \zeta (3)\,}$, the way there is for ${\displaystyle \scriptstyle \zeta (2n)\,}$ (with ${\displaystyle \scriptstyle n\,}$ a positive integer). However, with Plouffe's ${\displaystyle \scriptstyle S_{\pm }(n)\,}$ function[2][3], there is

${\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {\coth(\pi k)}{k^{3}}}-2\,S_{-}(3)={\frac {7}{180}}\pi ^{3}-2\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}},\,}$

where ${\displaystyle \scriptstyle \coth \,}$ is the hyperbolic cotangent function, and Plouffe's ${\displaystyle \scriptstyle S_{\pm }\,}$ function is

${\displaystyle S_{\pm }(n)\equiv \sum _{k=1}^{\infty }{\frac {1}{k^{n}\,(e^{2\pi k}\pm 1)}}.\,}$

## Decimal expansion of Apéry's constant

The decimal expansion of Apéry's constant is

${\displaystyle \zeta (3)=1.2020569031595942853997\ldots \,}$

giving the sequence of decimal digits (A002117)

{1, 2, 0, 2, 0, 5, 6, 9, 0, 3, 1, 5, 9, 5, 9, 4, 2, 8, 5, 3, 9, 9, 7, 3, 8, 1, 6, 1, 5, 1, 1, 4, 4, 9, 9, 9, 0, 7, 6, 4, 9, 8, 6, 2, 9, 2, 3, 4, 0, 4, 9, 8, 8, 8, 1, 7, 9, 2, 2, 7, 1, 5, 5, 5, 3, 4, 1, 8, 3, 8, 2, 0, 5, 7, 8, 6, 3, 1, 3, ...}

## Continued fraction expansion of Apéry's constant

The simple continued fraction expansion (aperiodic) of Apéry's constant is

${\displaystyle \zeta (3)=1\,+\,{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{18+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{\cdots }}}}}}}}}}}}}}}}}}\,}$

giving the sequence (A013631)

{1, 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, 1, 1, 1, 3, 1, 6, 1, 30, 1, 4, 1, 1, 4, 1, 3, 1, 2, 7, 1, 3, 1, 2, 2, 1, 16, 1, 1, 3, 3, 1, 2, 2, 1, 6, 1, 1, 1, 6, 1, 1, 4, 428, 5, 1, 1, 3, 1, 1, 11, 2, 4, 4, 5, 4, 1, ...}

2. For the definition of the ${\displaystyle \scriptstyle S_{\pm }(n)\,}$ function, see equation (94) in Jonathan Sondow and Eric W. Weisstein, Riemann Zeta Function, from MathWorld — A Wolfram Web Resource.