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

TL;DR

This paper investigates the asymptotic behavior of Ramanujan's third order mock theta function, proving conjectures about its series representation and deriving bounds using spectral theory of Maass forms, with applications to partition functions.

Contribution

It provides a power savings bound for Andrews' asymptotic formula, proves Andrews' conjecture on convergence, and introduces new formulas involving quadratic points.

Findings

Proved a power savings bound for the error in Andrews' formula.

Confirmed Andrews' conjecture that the series converges to the mock theta value.

Derived bounds for the error in Rademacher's partition function formula.

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 $$\alpha(n)= \sum_{c\leq\sqrt{n}} \psi(n)+O_\epsilon\left(n^\epsilon\right),$$ 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 which we derive 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.