In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group M and the j-function. The Fourier expansion of is given by

(Sloane's A000521), where and is the half-period ratio, and the dimensions of the first few irreducible representations of M are 1, 196883, 21296876, 842609326, ... (Sloane's A001379).

The first thing noticed by Conway and Guy (1979) was that the q-coefficient, 196884, is exactly one more than the smallest dimension of nontrivial representations of the M. In fact, it turns out that the Fourier coefficients of can be expressed as linear combinations of these dimensions with small coefficients as follows:

Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly ,there turn out to be yet more deep connections between the monster group and the j-function.

 

j-Function, Monster Group