Algebraic properties of the values of newform Dedekind sums

TL;DR

This paper investigates the algebraic structure of generalized Dedekind sums associated with Eisenstein series, revealing their lattice structure within certain number fields and extending classical identities.

Contribution

It establishes that the values of these generalized Dedekind sums form a full-rank lattice in a specific number field and generalizes Knopp's identity for classical Dedekind sums.

Findings

Values form a full-rank lattice in a number field

Generalization of Knopp's identity

Provides algebraic properties of Dedekind sums

Abstract

We study the image of a generalized Dedekind sum relating to the weight zero Eisenstein series $E_{\chi_1,\chi_2}$. We show that the image is a lattice of full rank inside a number field determined by the characters $\chi_1$ and $\chi_2$. We also give a generalization of Knopp's identity for the classical Dedekind sum.