Rademacher sums, moonshine and gravity

Understand

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expression- the first Rademacher sum - to verify its modular invariance.

• By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for all groups commensurable with the modular group.
• We use these Rademacher sums to illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators.
• We obtain a new characterization of the discrete groups of monstrous moonshine in terms of Rademacher sums, and we develop connections between Rademacher sums and a family of monstrous Lie algebras recently introduced by Carnahan.