Abstract
Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here, we solve the general problem of understanding Ramanujan’s definition explicitly for any weight \(\frac{1}{2}\) mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.
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Acknowledgements
The authors are grateful to Bruce Berndt, Minjoo Jang, and Steffen Löbrich for useful comments which improved the paper, as well as to Robert Rhoades for useful discussions related to the paper.
The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant agreement no. 335220-AQSER. The second author thanks the University of Cologne and the DFG for their generous support via the DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.
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Bringmann, K., Rolen, L. Radial limits of mock theta functions. Mathematical Sciences 2, 17 (2015). https://doi.org/10.1186/s40687-015-0035-8
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DOI: https://doi.org/10.1186/s40687-015-0035-8