Higher Weight Generalized Dedekind Sums

TL;DR

This paper introduces higher weight generalized Dedekind sums derived from period integrals of Eisenstein series, exploring their properties, reciprocity laws, and quantum modular behavior, supported by computational evidence for a generalized conjecture.

Contribution

It extends the theory of Dedekind sums to higher weights, develops new formulas, and investigates their modular and arithmetic properties with computational support.

Findings

Derived a finite sum formula for generalized Dedekind sums

Demonstrated their behavior as quantum modular forms

Provided computational evidence supporting a generalized conjecture

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.