Maass forms and the mock theta function $f(q)$

Let  

\[ f(q):=1+\sum_{n=1}^\infty \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots(1+q^n)^2}=:1+\sum_{n=1}^{\infty} \alpha(n)q^n, \]

be the well-known third order mock theta of Ramanujan. Part of the importance of the function $f(q)$ arises from the fact that the coefficients $\alpha(n)$ are related to a fundamental combinatorial statistic.  In particular, we have

\[ \alpha(n)=N_{\rm e}(n)-N_{\rm o}(n), \]

where these denote the number of partitions of even and odd rank respectively. In 1964, George Andrews proved an asymptotic formula of the form 

\[\alpha(n)= \sum_{c \leq\sqrt{n}} \psi(n)+O_\epsilon (n^\epsilon ),\]

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 a new estimate which we derive for the Fourier coefficients of such forms which gives a power savings in the spectral parameter as compared to results of Duke and Baruch-Mao.

 

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 Ahlgren--Andersen when $24n-23$ is squarefree.