Abstract:
We introduce Harmonic Maass forms and present some of their analytic properties.
We continue to define their related L-series by using Laplace transform, and prove
their functional equations. Our primary objective is to develop a converse theorem
for these L-series in both integral and non-integral weights. This became achievable
through the definition of the mentioned L-series on a broader class of test functions.
To illustrate the idea, we first present an outline of the special case of weakly holo-
morphic modular forms on SL2(Z) and then extend it to Harmonic Maass forms.
Subsequently, we consider an example of using the converse theorem.
