Continued Fractions and Hardy Sums

TL;DR

This paper introduces a new representation of Hardy sums as sums over partial quotients of specially defined continued fractions, linking them to classical Dedekind sums and demonstrating their dense distribution in the plane.

Contribution

It defines non-classical continued fractions and proves Hardy sums can be expressed as sums of their partial quotients, providing new insights into their properties.

Findings

Hardy sums can be represented as sums over non-classical continued fractions.

The graph of Hardy sums is dense in  imes .

New proof of the density of Hardy sums in the plane.

Abstract

The classical Dedekind sums $s(d, c)$ can be represented as sums over the partial quotients of the continued fraction expansion of the rational $\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of $\theta$-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we give a new proof that the graph of the Hardy sums is dense in $\mathbf{R} \times \mathbf{Z}$.