### Abstract

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.

Original language | English |
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Pages (from-to) | 2337-2349 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 144 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2016 Jun 1 |

Externally published | Yes |

### Keywords

- Eichler integral
- Mock theta function
- Quantum modular form
- Radial limit
- Ramanujan

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*144*(6), 2337-2349. https://doi.org/10.1090/proc/12907