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MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS

Published online by Cambridge University Press:  19 September 2013

AMANDA FOLSOM
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06520, USAamanda.folsom@yale.edu
KEN ONO
Affiliation:
Department of Mathematics, Emory University, Atlanta, GA 30322, USAono@mathcs.emory.edu
ROBERT C. RHOADES
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USArhoades@math.stanford.edurob.rhoades@gmail.com

Abstract

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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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
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/>.
Copyright
© The Author(s) 2013.

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