Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Abstract

Let rn denote the number of representations of a positive integer n as a sum of two squares, i.e., n=x12+x22, where x1 and x2 are integers. We study the behavior of the exponential sum twisted by rn over the arithmetic progressions ∑n∼Xn≡lmodqrneαnβ, where 0≠α∈ℝ, 01 is a large parameter, 1≤l≤q are integers, and l,q=1. We obtain the upper bounds in different situations.