On bilinear sums with modular square roots and applications II

TL;DR

This paper advances the understanding of bilinear sums involving modular square roots, improving bounds for large variable ranges and applying these results to enhance the large sieve for square moduli.

Contribution

It provides improved bounds for bilinear sums with modular square roots in large ranges and advances the large sieve for square moduli.

Findings

Enhanced bounds for bilinear sums with modular square roots

Progress on the large sieve for square moduli

Refined estimates for additive energies involving modular square roots

Abstract

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.