On the distribution of modular square roots of primes

TL;DR

This paper investigates how solutions to quadratic congruences involving primes and moduli are distributed, using advanced bounds on bilinear sums with modular square roots, blending previous approaches with new averaging techniques.

Contribution

It introduces new bounds on bilinear sums with modular square roots to analyze the distribution of solutions to quadratic congruences involving primes and moduli.

Findings

Derived bounds improve understanding of solution distribution

Unified previous approaches with new averaging techniques

Enhanced estimates for primes and moduli in quadratic congruences

Abstract

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.