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Journal of the London Mathematical Society Advance Access originally published online on February 13, 2008
Journal of the London Mathematical Society 2008 77(2):320-334; doi:10.1112/jlms/jdm119
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© 2008 London Mathematical Society

Duality involving the mock theta function f(q)

Amanda Folsom and Ken Ono

Department of Mathematics
University of Wisconsin
Madison WI 53706
USA
ono@math.wisc.edu

We show that the coefficients of Ramanujan's mock theta function f(q) are the first non-trivial coefficients of a canonical sequence of modular forms. This fact follows from a duality which equates coefficients of the holomorphic projections of certain weight 1/2 Maass forms with coefficients of certain weight 3/2 modular forms. This work depends on the theory of Poincaré series, and a modification of an argument of Goldfeld and Sarnak on Kloosterman–Selberg zeta functions.

Dedicated to Dorian Goldfeld on the occasion of his sixtieth birthday

2000 Mathematics Subject Classification 11F37, 33D15.

The authors thank the National Science Foundation for their generous support.The first author is supported by an NSF Postdoctoral Fellowship.

Received April 13, 2007; revised November 10, 2007; published online February 13, 2008.


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