Mock modular form

A mock theta function is one of a number of functions written down by Ramanujan with properties similar to those of theta functions of 1-dimensional lattices.

Zwegers showed in 2002 that mock theta functions can be written as the sum of a real analytic modular form of weight 1/2 and a function that behaves well at cusps, essentially disproving a longstanding conjecture of Ramanujan's that this was not possible.

References

Mock Theta Functions by Sander Pieter Zwegers