Completions and algebraic formulas for the coefficients of Ramanujan's mock theta functions

Abstract

We present completions of mock theta functions to harmonic weak Maass forms of weight $\frac{1}{2}$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $\frac{1}{2}$ that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.

Keywords: Harmonic weak Maass form; Mock theta function; Theta lift; Traces of singular moduli.