Uniform Distribution of Hardy Sums

TL;DR

This paper proves that Hardy sums, which are integer-valued analogs of Dedekind sums, are uniformly distributed modulo any integer m using spectral theory of Eisenstein series.

Contribution

It introduces a novel spectral approach to establish the uniform distribution of Hardy sums in modular arithmetic.

Findings

Hardy sums are uniformly distributed modulo m for all integers m > 1.

Spectral theory of Eisenstein series is effective in analyzing distribution properties of arithmetic sums.

The method provides a new perspective on the distribution of sums related to Dedekind sums.

Abstract

We employ the spectral theory of Eisenstein series to prove that the Hardy sums, integer-valued analogs of the classical Dedekind sums, are uniformly distributed in $\mathbf{Z} / m \mathbf{Z}$ for any integer $m > 1$.