A Short Proof of the Mock Theta Conjectures Using Maass Forms

Mock Theta Functions, Ranks, and Maass Forms

Computation of Harmonic Weak Maass Forms

Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan’s mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automo...

Coefficients of Harmonic Maass Forms

Harmonic Maass forms have recently been related to many different topics in number theory: Ramanujan’s mock theta functions, Dyson’s rank generating functions, Borcherds products, and central values and derivatives of quadratic twists of modular L-functions. Motivated by these connections, we obtain exact formulas for the coefficients of harmonic Maass forms of non-positive weight, and we obtai...

Almost a Century of Answering the Question: What Is a Mock Theta Function?

Quite a few famous and extraordinarily gifted mathematicians led lives that were tragically cut short. Ramanujan is certainly among them. While suffering from a fatal disease, he discovered what he called mock theta functions. Three months before his death in 1920 at the age of 32, he described them in a letter to Hardy that was written under difficulties and is in places very obscure. [30, p.3...

Exact Formulas for Coefficients of Jacobi Forms

In previous work, we introduced harmonic Maass-Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass-Jacobi-Poincaré series, as well as Zwegers’ real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass-Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this...