Definition:Apéry's Constant

Definition

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

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

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

• Value of Apéry's Constant

• Apéry's Theorem

• 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.

Sources

• 1979: Roger Apéry: Irrationalité de $\map \zeta 2$ et $\map \zeta 3$ (Astérisque Vol. 61: pp. 11 – 13)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,20205 69 \ldots$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 202 \, 056 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 20205 \, 6 \ldots$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Apéry's constant

• Weisstein, Eric W. "Apéry's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AperysConstant.html