Apéry's constant is defined as:

 

 

where ζ is the Riemann zeta function.

 

It has the approximate value 1.202056903...

 

It also arises in the following way: pick three positive integers at random, then the odds of them having no common divisor are 1 in 1.202056903...

 

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.

 

Reference

1. Apéry, R. "Irrationalité de ζ(2) et ζ(3)." Astérisque, 61: 11-13 (1979).