Nikolaos Diamantis

In a previous work (arXiv:2505.05574), a summation formula for harmonic Maass forms of polynomial growth was established. In this note, we use the theory of $L$-series of harmonic Maass forms to state and prove a summation formula for such forms without any restrictions on their growth. We deduce a summation formula for the partition function. We further employ the same theory to derive a result on classical modular forms, namely, a summation formula and the asymptotics of a Riesz sum attached to a cusp form.

We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.

We correct an error affecting Propositions 5.2 and Theorem 5.4 of [1]

Given the L-series of a half-integral weight cusp form, we construct polynomials behaving similarly to the classical period polynomial of an integral weight cusp form. We also define a lift of half-integral weight cusp forms to integral weight cusp forms that are compatible with the L-series of the respective forms.

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the “central L L -value” of the modular j j -invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this “ L L -value” as the value of an actual L L -series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L L -series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L L -values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L L -functions using test functions.

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.

We define an analogue of the Bol operator on spaces of weakly holomorphic modular forms of half-integral weight. We establish its main properties and relation with other objects.

Based on the theory ofL-series associated with weakly holomorphic modular forms in Diamantis et al. (L-series of harmonic Maass forms and a summation formula for harmonic lifts.arXiv:2107.12366), we derive explicit formulas for central values of derivatives ofL-series as integrals with limits inside the upper half-plane. This has computational advantages, already in the case of classical holomorphic cusp forms and, in the last section, we discuss computational aspects and explicit examples.