On roots of quadratic congruences

TL;DR

This paper improves bounds on the distribution of roots of quadratic congruences with prime moduli, focusing on Weyl linear forms for positive discriminants, advancing understanding in number theory.

Contribution

It provides a stronger estimate for the Weyl linear form in quadratic congruences with positive discriminants, extending previous results.

Findings

Enhanced bounds for Weyl linear forms with positive discriminant quadratic polynomials

Improved understanding of root distribution in quadratic congruences

Extension of previous estimates by Tóth and others

Abstract

The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for a special Weyl linear form. Duke, Friedlander and Iwaniec discovered a strong estimate for this Weyl linear form when the quadratic polynomial has negative discriminant. T\'oth established an analogous but weaker bound when the quadratic polynomial has positive discriminant. We obtain a stronger estimate for the Weyl linear form for quadratics of positive discriminants.