Maass forms and the mock theta function f(q)

Abstract

Let $f(q):=1+\sum _{n=1}^{\infty } \alpha (n)q^n$ be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form

$$\begin{aligned} \alpha (n)= \sum _{c\le \sqrt{n}} \psi (n)+O_\epsilon \left( n^\epsilon \right) , \end{aligned}$$

where $\psi (n)$ is an expression involving generalized Kloosterman sums and the I-Bessel function. Andrews conjectured that the series converges to $\alpha (n)$ when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews’ formula, and we also prove the second of these conjectures. Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on an average estimate for the Fourier coefficients of such forms which gives a power savings in the spectral parameter. As a further application of this result, we derive a formula which expresses $\alpha (n)$ with small error as a sum of exponential terms over imaginary quadratic points (this is similar in spirit to a recent result of Masri). We also obtain a bound for the size of the error term incurred by truncating Rademacher’s analytic formula for the ordinary partition function which improves a result of the first author and Andersen when $24n-23$ is squarefree.