A note on bilinear sums with modular square roots

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.