Apéry's constant

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

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

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

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

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

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

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

Decimal expansion of Apéry's constant

The decimal expansion of Apéry's constant is

$\zeta (3)=1.2020569031595942853997\ldots \,$

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

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

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, ...}

See also

• Zeta(2) = (pi^2)/6

External links

• Eric W. Weisstein, Apéry's Constant, from MathWorld — A Wolfram Web Resource.