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# Apéry's constant

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

^{[1]}

where is the Riemann zeta function and, in Euler's product, is the ^{th} prime.

It is unknown whether the constant is transcendental. It is also unknown whether there is a simple formula for , the way there is for (with a positive integer). However, with Plouffe's function^{[2]}^{[3]}, there is

where is the hyperbolic cotangent function, and Plouffe's function is

## Contents

## Decimal expansion of Apéry's constant

The decimal expansion of Apéry's constant is

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

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

## Notes

- ↑ For the cubes, see A000578.
- ↑ For the definition of the function, see equation (94) in Jonathan Sondow and Eric W. Weisstein, Riemann Zeta Function, from MathWorld — A Wolfram Web Resource.
- ↑ Simon Plouffe, "Identities Inspired from Ramanujan Notebooks II." Jul. 21, 1998. http://www.lacim.uqam.ca/~plouffe/identities.html.

## External links

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