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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Folsom-Ono grid contains only integers
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by Sander Zwegers PDF
Proc. Amer. Math. Soc. 137 (2009), 1579-1584 Request permission

Abstract:

In a recent paper, Folsom and Ono constructed a canonical sequence of weight 1/2 mock theta functions and a canonical sequence of weight 3/2 weakly holomorphic modular forms, both using Poincaré series. They show a remarkable symmetry in the coefficients of these functions and conjecture that all the coefficients are integers. We prove that this conjecture is true by giving an explicit construction for the weight 1/2 mock theta functions, using some results found by Guerzhoy.
References
  • Kathrin Bringmann and Ken Ono, The $f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266. MR 2231957, DOI 10.1007/s00222-005-0493-5
  • Amanda Folsom and Ken Ono, Duality involving the mock theta function $f(q)$, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 320–334. MR 2400394, DOI 10.1112/jlms/jdm119
  • —, Corrigendum: Duality involving the mock theta function $f(q)$, J. London Math. Soc., to appear.
  • P. Guerzhoy, Rationality of the Folsom–Ono grid, Proc. Amer. Math. Soc., accepted for publication.
  • David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
  • PARI/GP, version 2.3.1, Bordeaux, 2006, http://pari.math.u-bordeaux.fr/.
  • S. P. Zwegers, Mock $\theta$-functions and real analytic modular forms, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR 1874536, DOI 10.1090/conm/291/04907
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Additional Information
  • Sander Zwegers
  • Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
  • Email: sander.zwegers@ucd.ie
  • Received by editor(s): June 30, 2008
  • Received by editor(s) in revised form: July 2, 2008
  • Published electronically: November 18, 2008
  • Communicated by: Ken Ono
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1579-1584
  • MSC (2000): Primary 11F11, 11F27
  • DOI: https://doi.org/10.1090/S0002-9939-08-09684-6
  • MathSciNet review: 2470815