Apéry's constant ζ(3) is the sum 1/1³ + 1/2³ + 1/3³ + ⋯, the Riemann zeta function evaluated at s=3. Unlike ζ(2) = π²/6 (the Basel problem, solved by Euler), the value of ζ(3) has no known closed form. Its decimal expansion begins 1.20205690…
In 1978, Roger Apéry: a 64-year-old French mathematician: announced a proof that ζ(3) is irrational. His proof was elementary and unprecedented. The mathematical community was initially sceptical (one wrote "he must be nuts"), but the proof was verified and is now considered one of the great surprises of 20th-century mathematics.
Whether ζ(3) is transcendental, and whether ζ(5), ζ(7), ζ(9)… are even irrational, remains unknown. It appears in quantum electrodynamics and everywhere in physics where cubic corrections arise.