Transcendence of Apéry’s constant ζ(3)

Determine whether Apéry’s constant ζ(3) is transcendental; that is, ascertain whether ζ(3) is not a root of any non-zero polynomial with integer coefficients.

Background

Apéry’s constant ζ(3) is defined as the sum of the reciprocals of the positive cubes, ζ(3) = ∑_{n=1}^∞ 1/n^3, and has numerical value approximately 1.20205. In 1979, Roger Apéry proved that ζ(3) is irrational, establishing a landmark result about special values of the Riemann zeta function.

Within the paper’s exploration of approximating mathematical constants using Minecraft, ζ(3) is connected to a probabilistic interpretation: the reciprocal 1/ζ(3) equals the probability that three uniformly random positive integers are coprime. Despite extensive study and multiple series and integral representations, the transcendence of ζ(3) remains unresolved.