Mock Theta Functions and Quantum Modular Forms

Abstract

Ramanujan's last letter to Hardy concerns the asymptotic properties of modular forms and his 'mock theta functions'. For the mock theta function f (q), Ramanujan claims that as q approaches an even-order 2k root of unity, we have f (q) − (−1) k (1 − q)(1 − q 3 )(1 − q 5 ) · · · (1 − 2q + 2q 4 − · · ·) = O(1).We prove Ramanujan's claim as a special case of a more general result. The implied constants in Ramanujan's claim are not mysterious. They arise in Zagier's theory of 'quantum modular forms'. We provide explicit closed expressions for these 'radial limits' as values of a 'quantum' q-hypergeometric function which underlies a new relationship between Dyson's rank mock theta function and the Andrews-Garvan crank modular form. Along these lines, we show that the Rogers-Fine false ϑ-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms. 2010 Mathematics Subject Classification: 11F99 (primary); 11F37, 33D15 (secondary) Overview In his 1920 deathbed letter to Hardy, Ramanujan gave examples of 17 curious q-series he referred to as 'mock theta functions' [11]. In the decades following Ramanujan's death, mathematicians were unable to determine how these functions fit into the theory of modular forms, despite c The Author(s) 2013. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence <http://creativecommons.org/licenses/by/3.0/>. A. Folsom, K. Ono, and R. C. Rhoades 2 their rather ubiquitous nature. Finally, the 2002 Ph.D. thesis of Zwegers [45] showed that while the mock theta functions were not modular, they could be 'completed' to produce real analytic vector-valued modular forms. Zwegers's breakthrough catalyzed the development of the overarching theory of 'weak Maass forms' by Bringmann, Ono, and collaborators [15, 16, 35, 42]. Ramanujan's mock theta functions, it turns out, are examples of 'holomorphic parts' of weak Maass forms, originally defined by Bruinier and Funke [18].While the theory of weak Maass forms has led to a flood of applications in many disparate areas of mathematics (see [35,42] and references therein), it is still not the case that we fully understand the deeper framework surrounding the contents of Ramanujan's last letter to Hardy. Here, we revisit Ramanujan's original claims and motivations. His last letter summarizes asymptotic properties near roots of unity of modular 'Eulerian' series. Ramanujan asks whether other Eulerian series with similar asymptotics are necessarily the sum of a modular theta function and a function which is O(1) at all roots of unity. He writes: 'The answer is it is not necessarily so . . . I have not proved rigorously that it is not necessarily so . . . But I have constructed a number of examples . . . '. In fact, Ramanujan's sole example and claim pertain to his third-order mock theta function f (q). CLAIM (Ramanujan [11]). As q approaches an even-order 2k root of unity, we haveHere, we prove (in ...

Asymptotic formulas for coefficients of inverse theta functions

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