Equidistribution of generalized Dedekind sums and exponential sums

TL;DR

This paper establishes the equidistribution of generalized Dedekind sums by linking them to exponential sums and applying Weil bounds, extending classical results without relying on modular forms.

Contribution

It introduces a new exponential sum associated with generalized Dedekind sums and proves their equidistribution using Weil bounds and cohomological methods.

Findings

We obtain Weil bounds for the exponential sums related to generalized Dedekind sums.

The sums satisfy Weyl's equidistribution criterion, confirming their uniform distribution.

Special case recovers classical Dedekind sum equidistribution without modular form assumptions.

Abstract

For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this exponential sum using the cocycle property of the Todd series of 2d cones and the nonsingular cone decomposition along with the continued fraction of q/p. Its Weil bound is given for the modulus q, applying the purity theorem of the cohomology of the related l-adic sheaf due to Denef and Loeser. The Weil type bound of Denef and Loeser fulfills the Weyl's equidistribution criterion for R(i,j)q^{i+j-2} s_{ij}(p,q). As a special case, we recover the equidistribution result of the classical Dedekind sums multiplied by 12 not using the modular weight of the Dedekind's \eta(\tau).