Bounding the exponential sum on squares of some sifted sequences

TL;DR

This paper establishes bounds on exponential sums over squares of sifted sequences, specifically odd primitive Gaussian integers, extending results beyond sums of two squares.

Contribution

It provides new bounds on exponential sums involving Gaussian integers, generalizing previous results on sums of two squares.

Findings

Bound on exponential sum: $S(eta;N)/ (N/\sqrt{\log N}) \ll N^\epsilon (q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8})$

Results extend to more general sequences beyond sums of two squares

Applicable to sequences characterized by Gaussian integers and their quadratic exponential sums.

Abstract

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le N}b(n)e(n^2\alpha)$ satisfies \begin{equation*} \frac{S(\alpha;N)}{N/\sqrt{\log N}} \ll N^\epsilon (q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|\alpha - a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.