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

L-values of harmonic Maass forms

“…The geometric interpretation of the expression (1.3) established in [8], combined with the systematic approach to L-series of harmonic Maass forms of [11] that we apply here, suggests a deeper arithmetic meaning of the L-values considered which is worth studying further.…”

“…Our work aims to begin a more systematic study of modular L-values when there is exponential growth at the cusps. Recently, Lee, Raji, and the authors developed [11] a new theory of L-series in such cases. The key idea was to use a distributional definition, rather than think of L-series as functions on C. This distributional definition allowed the development of structures that are critically important for holomorphic modular forms, e.g.…”

“…functional equations, converse theorems, Voronoi-type summation formulas. Here, we apply the approach of [11] to study periods.…”

“…As pointed out in [8] and [1], the reason that the above explicit formulas could be considered as L-values only by analogy is that, at the time, there was no systematic construction of L-series for harmonic Maass forms. Recently, however, the authors jointly with M. Lee and W. Raji have defined and studied actual L-series for general harmonic Maass forms ( [11]). In this paper, we will use the theory of [11] to interpret the expressions established in [8] and [1] in the framework of a properly defined L-series.…”

“…Recently, however, the authors jointly with M. Lee and W. Raji have defined and studied actual L-series for general harmonic Maass forms ( [11]). In this paper, we will use the theory of [11] to interpret the expressions established in [8] and [1] in the framework of a properly defined L-series. We will further extend those explicit expressions beyond the "central value" to include, in particular, all integer points.…”