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Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

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  • Li Ma
  • Xiaofei Yan
  • Tianping Zhang

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∈℠, 0 1 is a large parameter, 1≤l≤q are integers, and l,q=1. We obtain the upper bounds in different situations.

Suggested Citation

  • Li Ma & Xiaofei Yan & Tianping Zhang, 2021. "Resonance between the Representation Function and Exponential Functions over Arithemetic Progression," Journal of Mathematics, Hindawi, vol. 2021, pages 1-10, February.
    • Handle: RePEc:hin:jjmath:6616348
    DOI: 10.1155/2021/6616348
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