Generalized Dedekind sums and equidistribution mod 1

TL;DR

This paper introduces a generalized concept of Dedekind sums, called Dedekind symbols, for broader classes of lattices, and proves their equidistribution mod 1, extending classical results and connecting to modular forms and exponential sums.

Contribution

It defines Dedekind symbols for any non-cocompact lattice in SL(2,R) and establishes their equidistribution mod 1, generalizing classical Dedekind sums.

Findings

Dedekind symbols are well-defined for non-cocompact lattices.

Dedekind symbols exhibit equidistribution mod 1.

Connection to Kloosterman sums extends classical results.

Abstract

Dedekind sums are well-studied arithmetic sums, with values uniformly distributed on the unit interval. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of $SL(2,\mathbb{Z})$. We present a compatible notion of Dedekind sums, which we name Dedekind symbols, for any non-cocompact lattice $\Gamma<SL(2,\mathbb{R})$, and prove the corresponding equidistribution mod 1 result. The latter part builds up on a paper of Vardi, who first connected exponential sums of Dedekind sums to Kloosterman sums.