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- W2122332634 abstract "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 $$begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )cdots (1- 2q+ 2{q}^{4} - cdots )= O(1).end{eqnarray*}$$ 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." @default.
- W2122332634 created "2016-06-24" @default.
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- W2122332634 date "2013-01-01" @default.
- W2122332634 modified "2023-10-16" @default.
- W2122332634 title "MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS" @default.
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- W2122332634 doi "https://doi.org/10.1017/fmp.2013.3" @default.
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