# Harmonic Analysis on the Positive Rationals. Computation of Character Sums

## TL;DR

This paper explores harmonic analysis techniques on positive rationals, focusing on representing rationals through specific product and quotient forms, and introduces a novel two-dimensional representation of rational pairs.

## Contribution

It provides a detailed account of representing rationals via product/quotient forms and introduces a new simultaneous two-dimensional rational representation.

## Key findings

- Representation of rationals using $(an+b)/(An+B)$ forms
- A non-intuitive two-dimensional rational pair representation
- Insights into harmonic analysis on positive rationals

## Abstract

The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive simultaneous ( i.e. two dimensional ) representation of pairs of rationals.