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A framework for modular properties of false theta functions

  • Kathrin BringmannEmail author
  • Caner Nazaroglu
Research
  • 26 Downloads

Abstract

False theta functions closely resemble ordinary theta functions; however, they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the circle method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.

Keywords

Circle method Completions False theta functions Mock theta functions Modular forms 

Mathematics Subject Classification

11F12 11F20 11F27 11F30 11F37 11F50 11P82 

Notes

Acknowlegements

The authors thank Chris Jennings-Shaffer for helpful comments on an earlier version of the paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CologneCologneGermany

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