Apéry's constant arises as follows: pick three positive integers at random; then the odds of them having no common divisor are 1 in 1.202056... In 1979, the French mathematician Roger Apéry (1916–1994) stunned the mathematical world with a proof that this number is irrational.1 Whether it is a transcendental number remains an open question. Apéry's constant is also ζ(3), where ζ is the Riemann zeta function.
1. Apéry, R. "Irrationalité de ζ(2) et ζ(3)." Astérisque, 61: 11-13 (1979).