Number Theory Seminar

Maass forms and the mock theta function f(q)

Alexander Dunn, Department of Mathematics, Caltech,

Let f(q) be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula for the coefficients of f(q) as a finite sum of 1/2-integral weight Kloosterman sums and $I$-Bessel functions with O(n^{\varepsilon}) error term.

Confirming a conjecture of Andrews, Bringmann and Ono proved in 2009 that Andrew's formula converges when extended to an infinite sum. We obtain a power savings bound for the error in Andrews' formula using the spectral theory of Maass forms of half-integral weight. We also confirm a conjecture of Andrews about the absolute convergence of this series.

We would like to highlight this as a nice connection between integer partitions and spectral theory of automorphic forms.

This is a joint work with Scott Ahlgren.

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