Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Abstract

Let $r\left(n\right)$ denote the number of representations of a positive integer $n$ as a sum of two squares, i.e., $n=x_1^2+x_2^2$ , where $x_1$ and $x_2$ are integers. We study the behavior of the exponential sum twisted by $r\left(n\right)$ over the arithmetic progressions ${\displaystyle {\sum }_{\begin{array}{c}n\sim X\\ n\equiv l\mathrm{mod}q\end{array}}r\left(n\right)e\left(\alpha n^{\beta }\right)},$ where $0\ne \alpha \in ℝ$ , $0<\beta <1$ , $e\left(x\right)=e^{2\pi ix}$ , and $n\sim X$ means $X<n\le 2X$ . Here, $X>1$ is a large parameter, $1\le l\le q$ are integers, and $\left(l,q\right)=1$ . We obtain the upper bounds in different situations.