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
We resolve a question of Kac and explain the automorphic properties of characters due to Kac-Wakimoto pertaining to s`.mjn/^highest weight modules, for n 1. We prove that the Kac-Wakimoto characters are essentially holomorphic parts of certain generalizations of harmonic weak Maass forms which we call "almost harmonic Maass forms". Using a new approach, this generalizes prior work of the first author and Ono, and the authors, both of which treat only the case n D 1. We also provide an explicit asymptotic expansion for the characters.
We construct an infinite family of quantum modular forms from combinatorial rank "moment" generating functions for strongly unimodal sequences. The first member of this family is Kontsevich's "strange" function studied by Zagier. These results rely upon the theory of mock Jacobi forms. As a corollary, we exploit the quantum and mock modular properties of these combinatorial functions in order to obtain asymptotic expansions. n j−1 n+1 j=1 with n even. Attached to strongly unimodal sequences is a notion of rank, analogous to the well-known notion of the rank of an integer partition. For more on partition ranks, see for example original works in [Ramanujan 1919; Dyson 1944; Atkin and Swinnerton-Dyer 1954], and the more recent joint work of [Bringmann and Ono 2010] related to mock modular forms. The rank of a strongly unimodal sequence is equal to s − 2k + 1, the number of terms after the maximal term minus the number of terms that precede it. For example, there are six strongly unimodal sequences of size 5: {5},
Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.
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