# Definition:Apéry's Constant

## Definition

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:

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

## 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.