Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.
Before the work of Zwegers, the orders of known mock theta functions included Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.
This changed in 2001 when Zwegers discovered the relation with non-holomorphic modular forms, Lerch sums, and indefinite theta series.
Zwegers's fundamental result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2.
In particular, like modular forms, mock theta functions all lie in certain explicit finite-dimensional spaces, which reduces the long and hard proofs of many identities between them to routine linear algebra.
As further applications of Zwegers's ideas, Kathrin Bringmann and Ken Ono showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms[6] and showed that the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by George Andrews[7] and Leila Dragonette[8] converges to the coefficients.
Don Zagier[11] defines a mock theta function as a rational power of q = e2πi𝜏 times a mock modular form of weight 1/2 whose shadow is a theta series of the form for a positive rational κ and an odd periodic function ε.
The space of mock modular forms (of given weight and group) whose growth is bounded by some fixed exponential function at cusps is finite-dimensional.
Appell–Lerch sums, a generalization of Lambert series, were first studied by Paul Émile Appell[12] and Mathias Lerch.
Zwegers used this idea to express mock theta functions as Fourier coefficients of meromorphic Jacobi forms.
[5] The latter proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appell–Lerch sums.
[26] In his lost notebook he stated some further identities relating these functions, equivalent to the mock theta conjectures,[27] that were proved by Hickerson.
[28] Andrews[14] found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight 1/2.
Ramanujan[4] wrote down seven mock theta functions of order 6 in his lost notebook, and stated 11 identities between them, which were proved by Andrews and Hickerson.
[14] Hickerson[15] found representations of many of these functions as the quotients of indefinite theta series by modular forms of weight 1/2.
They found five linear relations involving them, and expressed four of the functions as Appell–Lerch sums, and described their transformations under the modular group.
Ramanujan[34] listed four order-10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by Choi.