Two Geometric Interpretations of Hardy Sums

TL;DR

This paper explores two geometric interpretations of Hardy sums, providing explicit formulas for lattice points in triangles with scaled lattices and establishing a reciprocity law through geometric and intersection number perspectives.

Contribution

It introduces a second geometric interpretation of Hardy sums as intersection numbers and proves a generalized reciprocity law using elementary geometric arguments.

Findings

Explicit formula for lattice points in triangles with (2Z)^2 lattice

Geometric interpretation of Hardy sums as intersection numbers

Proof of a generalized reciprocity law for Hardy sums

Abstract

The problem of finding the number of lattice points in a triangle has a classical solution if the lattice is $\mathbf{Z}^2$ and the vertices of the triangle have integer valued coordinates. We consider what happens when we replace the lattice by $(2 \mathbf{Z})^2$ instead and give an explicit formula for the number of lattice points inside a triangle in terms of Hardy sums. Moreover, we give a second geometric interpretation of the Hardy sums as signed intersection numbers with a certain oriented net of geodesics. Using this geometric realization, we prove a generalized reciprocity law for Hardy sums by an elementary argument.