Definition:Apéry's Constant

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Apéry's constant is the value of the infinite sum:

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

where $\zeta$ denotes the Riemann zeta function.

Its approximate value is given by:

$\map \zeta 3 \approx 1 \cdotp 20205 \, 69031 \, 59594 \, 28539 \, 97381 \, 61511 \, 44999 \, 07649 \, 86292 \ldots$

This sequence is A002117 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

  • Results about Apéry's constant can be found here.

Source of Name

This entry was named for Roger Apéry.

Historical Note

Apéry's constant was first investigated by Leonhard Paul Euler, who tried but failed to calculate its value.

Its precise value is still unknown.

However, in $1979$ Roger Apéry published a proof that it is irrational.

The statement of this fact is now known as Apéry's Theorem, and the constant itself is known as Apéry's constant.