Mock modular forms and quantum modular forms

In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples f(q), Ramanujan claimed that there is a collection {G_j} of modular forms such that for each root of unity ζ, there is a j such that (f(q) − G_j (q)) = O(1). Moreover, Ramanujan claimed that this collection must have size larger than 1. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size 1. These results arise within the context of Zagier’s theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for “Ramanujan’s radial limits” as L-values.