# A congruence sum and rational approximations

## TL;DR

This paper introduces a reciprocity formula for a two-variable sum constrained by a linear congruence, linking it to rational approximation quality and providing insights into number theory.

## Contribution

It presents a new reciprocity formula for a specific sum and connects it to rational approximation, offering a novel perspective in number theory.

## Key findings

- Reciprocity formula for the two-variable sum
- Sum measures how well a rational number can be approximated from below
- Reciprocity formula derived from rational approximation properties

## Abstract

We give a reciprocity formula for a two-variable sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is approximable from below and show that the reciprocity formula is a simple consequence of this fact.