A note on bilinear sums with modular square roots

TL;DR

This paper extends previous work on bilinear sums over finite fields to include sums involving modular square roots, providing new bounds and insights for these sums.

Contribution

It introduces an analogous result for bilinear sums with modular square roots, building on prior methods and reviewing related recent research.

Findings

Established bounds for bilinear sums involving modular square roots.

Extended previous methods to the case s=1/2 in bilinear sums.

Reviewed recent related work on bilinear sums.

Abstract

Bag and Shparlinski \cite{BaSh} considered bilinear sums of terms of the form $e_p(axy^s)$, where $p$ is a prime, $a$ is an integer coprime to $p$, $s$ is an integer, $x$ runs over a subset of $\mathbb{F}_p^{\ast}$ and $y$ runs over an interval. Closely following their method, we establish an analogous result for the case when $s=1/2$ ($y^{1/2}$ being a modular square root of $y$ modulo $p$, if existent). A part of this note is devoted to reviewing our recent works on related bilinear sums.