The Density of Elliptic Dedekind Sums

TL;DR

This paper proves that for lattices with real j-invariant, normalized elliptic Dedekind sums are dense in the real numbers, extending previous results to a broader class of complex lattices.

Contribution

It establishes the density of elliptic Dedekind sums for lattices with real j-invariant, generalizing earlier density results to complex lattices.

Findings

Normalized elliptic Dedekind sums are dense in the real numbers for lattices with real j-invariant.

The proof adapts Kohnen's recent methods to elliptic Dedekind sums.

Extends previous density results from Euclidean imaginary quadratic rings to more general lattices.

Abstract

Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.