On bilinear sums with modular square roots and applications III

TL;DR

This paper advances the analysis of bilinear sums with modular square roots, specifically addressing prime square moduli by modifying previous methods to achieve new bounds through quadratic Gauss sum restrictions.

Contribution

It introduces a modified approach to handle prime square moduli, enabling progress where previous methods failed, by restricting quadratic Gauss sums to residue classes.

Findings

Achieved new bounds for bilinear sums with prime square moduli.

Introduced a method to restrict quadratic Gauss sums for better cancellations.

Extended previous results to a new class of moduli.

Abstract

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.