L-series of harmonic Maass forms and a summation formula for harmonic lifts

TL;DR

This paper introduces L-series for harmonic Maass forms, proves their functional equations, and establishes a summation formula for harmonic lifts of cusp forms, advancing understanding of their analytic properties.

Contribution

It is the first to define L-series for harmonic Maass forms, prove their functional equations, and derive a summation formula for harmonic lifts.

Findings

L-series associated with harmonic Maass forms satisfy functional equations

Converse theorems for these L-series are established

A summation formula for the holomorphic part of harmonic lifts is proved

Abstract

We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.