Abstract
Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here, we solve the general problem of understanding Ramanujan’s definition explicitly for any weight \(\frac{1}{2}\) mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.
Highlights
1 Background In this paper, we study a general problem of Rhoades [21] on the nature of mock theta functions near the cusps, inspired by important examples of Ramanujan [3], Folsom, Ono, and Rhoades [11], and others
Before stating the precise question of Rhoades, we briefly recall the history of the mock theta functions
Thanks to the pioneering work of Zwegers [28] and Bruinier and Funke [6], we understand this structure in terms of so-called harmonic Maass forms
Summary
We study a general problem of Rhoades [21] on the nature of mock theta functions near the cusps, inspired by important examples of Ramanujan [3], Folsom, Ono, and Rhoades [11], and others (for example, see [1, 12, 27]). Ramanujan considered the q-series defined by: qn f (q) := n≥0 (−q)2n , Bringmann and Rolen Mathematical Sciences (2015) 2:17 where throughout q := e2πiτ with τ ∈ H and, for n ∈ N0 ∪ {∞}, (a)n := (a; q)n := He noticed that all of his examples look like classical modular forms (or theta functions, as he called them) as one approaches roots of unity. 6 for a simple example which illustrates the theorem As another application of Theorem 1.1, we recall that our question is closely related to a new type of nearly modular object known as a quantum modular form We conclude in Sect. 6” with an illuminating example and some concluding discussion
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