Resonance between the Representation Function and Exponential Functions over Arithemetic Progression - Li Ma (Journal of Mathematics, 2021)
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, 0<β<1, ex=e2πix, and n∼X means X<n≤2X. Here, X>1 is a large parameter, 1≤l≤q are integers, and l,q=1. We obtain the upper bounds in different situations.