A Shifted Sum for the Congruent Number Problem

TL;DR

This paper introduces a shifted convolution sum whose growth rate depends on whether a number is a congruent number, offering a new approach to studying the longstanding Congruent Number Problem.

Contribution

It proposes a novel shifted convolution sum parametrized by squarefree numbers, linking its asymptotic behavior to the congruent number property.

Findings

Series growth depends on congruent number status.

Provides a new analytical tool for the Congruent Number Problem.

Opens new research directions in number theory.

Abstract

We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$. The asymptotic growth of this series depends explicitly on whether or not $t$ is a \emph{congruent number}, an integer that is the area of a rational right triangle. This series presents a new avenue of inquiry for The Congruent Number Problem.