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Mock theta functions and quantum modular forms. (English) Zbl 1294.11083

Summary: 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 \[ f(q) - (-1)^k(1 - q)(1 - q^3)(1 - q^5)\cdots(1 - 2q + 2q^4 - \cdots) = O(1). \] 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 \(\vartheta\)-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

MSC:

11F99 Discontinuous groups and automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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