Apéry's Constant ζ(3)

Undergraduate

Definition

Apéry's constant is ζ(3) from the Riemann zeta function. Roger Apéry proved its irrationality in 1978.

ζ(3) = ∑_n=1^∈fty (1)/(n³) = 1 + (1)/(8) + (1)/(27) + ... ≈ 1.2020569...

Definition of Apéry's constant

ζ(3) = (5)/(2) ∑_n=1^∈fty \frac(-1)^n-1n³ \binom2nn

Apéry's series

Example 1

Find the sum of first 4 terms of ζ(3).

Discovered by: Roger Apéry (1978)

Proved irrationality of ζ(3) after 200+ years, shocking the mathematical community.

Zeta function research

Quantum electrodynamics

Bose gas