General Bilinear Forms In The Jacobi Symbol Over Hyperbolic Regions

TL;DR

This paper investigates the behavior of averages of the Jacobi quadratic symbol over hyperbolic regions, demonstrating cancellation effects when summing over square-free integers away from axes.

Contribution

It introduces new results on the cancellation of Jacobi symbol averages in hyperbolic regions, focusing on square-free integers and their distribution.

Findings

Averages of the Jacobi symbol show cancellation in specified regions.

Cancellation occurs when summing over square-free integers.

Results contribute to understanding quadratic symbol distributions in number theory.

Abstract

We study averages involving the Jacobi quadratic symbol $(\frac{n}{m})$ in regions where the product $mn$ is bounded by a large parameter. We show that these averages exhibit cancellation whenever the summation is restricted to square-free integers bounded away from the axes.