Abstract
We present completions of mock theta functions to harmonic weak Maass forms of weight nicefrac {1}{2} and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight nicefrac {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.
Highlights
Mock theta functions first appeared in Ramanujan’s last letter to his friend Hardy in 1920
Ramanujan did not give any definition of what a mock theta function should be, but listed 17 examples, divided into four groups of orders 3, 5, 7 and 10, respectively, given as q-hypergeometric series, and stated various identities between them and some analytical properties
The four mock theta functions of order 3 that Ramanujan defined in his letter are
Summary
Mock theta functions first appeared in Ramanujan’s last letter to his friend Hardy in 1920. Though Ramanujan had not explained what the order of a mock theta function should be, it turned out that the order is related to the level of the corresponding Maass form We will present such completions to a harmonic weak Mass form of weight 1/2 for 22 different mock theta functions of orders 2, 3, 6 and 8. By writing the harmonic weak Maass form of weight 1/2 containing the mock theta functions as the Millson theta lift of a suitable weakly holomorphic modular form, we can derive finite algebraic formulas for the coefficients of the considered mock theta functions in terms of traces of singular moduli. [11] where they have been proven in more detail
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