On three theorems of Folsom, Ono and Rhoades
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Abstract:
In his deathbed letter to G. H. Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotic matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to Dyson’s rank (mock theta) function and the Andrews–Garvan crank (modular) function — the match with explicit formulae for implied $O(1)$ constants. In this note we give another elementary proof of Ramanujan’s original claim and outline some heuristics which may be useful for obtaining a new proof of the general Folsom–Ono–Rhoades theorem.References
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Additional Information
- Wadim Zudilin
- Affiliation: School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia
- Email: wadim.zudilin@newcastle.edu.au
- Received by editor(s): September 11, 2013
- Published electronically: November 4, 2014
- Additional Notes: The author was supported by the Australian Research Council.
- Communicated by: Ken Ono
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 1471-1476
- MSC (2010): Primary 11F03; Secondary 11P84, 33D15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12364-1
- MathSciNet review: 3314062