Abstract

Ramanujan's 1920 last letter to Hardy contains seventeen examples of mock theta functions, including the famous third order function f(q), which resemble modular forms of weight 1/2. This list has since been expanded by individuals such as Gordon and McIntosh and the McKay--Thompson series arising in moonshine theory. In 1964, Andrews---improving on a result of Dragonette---gave an asymptotic formula for the coefficients of f(q) and conjectured an exact formula: a conditionally convergent series that closely resembles the Hardy--Ramanujan--Rademacher formula for the partition function. A key breakthrough came in 2001, when Zwegers wrote f(q) as the holomorphic component of a vector-valued harmonic Maass form for the Weil representation. Building upon this work, Bringmann and Ono proved Andrews' conjecture in 2006. In 2017, Bruinier and Schwagenscheidt gave another formula for the coefficients of f(q) as the algebraic trace of an eta quotient. We generalize these results for thirty-nine mock theta functions. We give an exact formula for each of their coefficients: both as a series resembling that of Andrews and as an algebraic trace of a Poincaré series. Along the way, we include five additional completions of mock theta functions as vector-valued harmonic Maass forms. We also give a general calculation for the coefficients of forms that arise as the Millson theta lift of a Poincaré series. Finally, we prove a general theorem bounding sums of Kloosterman sums for the Weil representation attached to a lattice of odd rank.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

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