Apéry's Constant

ζ(3) is 1 + 1/8 + 1/27 + 1/64 + ... = 1.2020569031595943. Euler solved the Basel problem in 1735, showing the sum of reciprocal squares is π²/6, and then found closed forms for every even power — but the odd ones defeated him and everyone since. There is still no known expression for ζ(3) in terms of π and elementary operations, and most experts doubt one exists.

The breakthrough came from an unexpected direction. In June 1978, at the Journées Arithmétiques in Marseille, the 61-year-old Roger Apéry gave a talk in French announcing that ζ(3) is irrational. The audience was openly sceptical — the methods looked too elementary and the identities he wrote down seemed to come from nowhere — and it took several mathematicians two months of work to verify every step. The result stands, and no one has since managed to prove the same for ζ(5) or any other odd value, though Wadim Zudilin showed in 2001 that at least one of ζ(5), ζ(7), ζ(9), ζ(11) must be irrational. ζ(3) also appears in physics, in the electron's gyromagnetic ratio corrections and in the Debye model of heat capacity.