Rademacher Sums and Rademacher Series
Miranda C. N. Cheng, John F. R. Duncan
TL;DR
The paper develops and analyzes Rademacher sums and their regularisation for arbitrary weights, clarifying when they yield modular or mock modular forms and how their shadows are determined. It introduces Rademacher series as a coefficient toolkit and proves dualities that connect data in dual weights, enabling explicit Fourier expansions for classical modular objects and moonshine-related functions. Moonshine is recast in a unified framework: monstrous, Mathieu, and umbral McKay–Thompson series are expressed as Rademacher sums or vector-valued variants, linking finite groups to mock modular forms and enabling a common structural viewpoint via Zagier-type dualities. The discussion culminates with physical interpretations in AdS/CFT, where the Rademacher construction aligns with gravity path integrals and saddle-point decompositions, highlighting the relevance of these sums to quantum gravity and string-theoretic contexts.
Abstract
We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics.