In mathematics, Apéry's theorem states that that ζ(3) is an irrational number where ζ denotes the Riemann zeta function.

ζ(3) is the infinite series of the reciprocals of the cube numbers (the cubes of the positive integers). This result was proved in 1977 by the French mathematician Roger Apéry (1916 - 1994).

The original proof was quite complex and hard to grasp. Subsequently quite short proofs have been found, using Legendre polynomials.

The result has remained quite isolated: little is known about ζ(n) for other odd numbers n.