A survey of classical mock theta functions

Abstract

In his last letter to Hardy, Ramanujan definned 17 functionsM(q), {pipe} q{pipe}< 1, which he called mock θ -functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an exponential singularity, there is a θ -function Tζ (q) with M(q)-Tζ (q)=O(1). Since then, other functions have been found which possess this property. We list various linear relations between these functions and develop their transformation laws under the modular group. We show that each mock θ -function is related to a member of a universal family (mock θ -conjectures). In recent years the infject has received new impetus and importance through a strong connection with the theory of Maass forms. The final section of this survey provides some brief remarks concerning these new developments. © Springer Science+Business Media, LLC 2012.