Higher Weight Generalized Dedekind Sums

Abstract

Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we explore a variety of properties of these generalized Dedekind sums: we develop a finite sum formula, demonstrate their behavior as quantum modular forms, provide a Fricke reciprocity law, and characterize analytic and arithmetic aspects of their image. Particularly, for the arithmetic aspect of the image, we generalize an existing conjecture to the higher weight case and provide significant computational evidence to support this generalized conjecture.