Abstract
AbstractRamanujan’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.
Highlights
Introduction and statement of resultsRamanujan’s enigmatic last letter to Hardy [11] gave tantalizing hints of his theory of mock theta functions
We provide an explicit closed formula for the implied constants in Ramanujan’s claim, and show that they are 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, two of the most studied q-series in the theory of partitions
Thanks to Zwegers [44, 45], it is known that these functions are specializations of nonholomorphic Jacobi forms
Summary
Ramanujan’s enigmatic last letter to Hardy [11] gave tantalizing hints of his theory of mock theta functions. U(w; q) play in Theorem 1.2, it is natural to ask about the more general relationship between mock theta functions and quantum modular forms. To this end, we seek q-hypergeometric series related to mock theta functions which are defined on both H+ and H−. The second equalities in (1.10) and (1.11) are only valid for |q| < 1 These specializations satisfy the following nice properties often associated to quantum modular forms: convergence in H±, a modular transformation law, and asymptotic expansions which are generating functions for values of L-functions. For h/k ∈ Qa,b, as t → 0+, we have
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