Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Abstract

Let <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>r</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>n</mi> </mrow> </mfenced> </math> denote the number of representations of a positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>n</mi> </math> as a sum of two squares, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>n</mi> <mo>=</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> </math> , where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> are integers. We study the behavior of the exponential sum twisted by <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>r</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>n</mi> </mrow> </mfenced> </math> over the arithmetic progressions <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mstyle displaystyle="true"> <msub> <mrow> <mo stretchy="false">∑</mo> </mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>n</mi> <mo>∼</mo> <mi>X</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>n</mi> <mo>≡</mo> <mi>l</mi> <mi mathvariant="normal">mod</mi> <mi>q</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </msub> <mrow> <mi>r</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>n</mi> </mrow> </mfenced> <mi>e</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>α</mi> <msup> <mrow> <mi>n</mi> </mrow> <mrow> <mi>β</mi> </mrow> </msup> </mrow> </mfenced> </mrow> </mstyle> <mo>,</mo> </math> where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mn>0</mn> <mo>≠</mo> <mi>α</mi> <mo>∈</mo> <mi>ℝ</mi> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mn>0</mn> <mo>&lt;</mo> <mi>β</mi> <mo>&lt;</mo> <mn>1</mn> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>e</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>=</mo> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>x</mi> </mrow> </msup> </math> , and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>n</mi> <mo>∼</mo> <mi>X</mi> </math> means <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <mi>X</mi> <mo>&lt;</mo> <mi>n</mi> <mo>≤</mo> <mn>2</mn> <mi>X</mi> </math> . Here, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <mi>X</mi> <mo>&gt;</mo> <mn>1</mn> </math> is a large parameter, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14"> <mn>1</mn> <mo>≤</mo> <mi>l</mi> <mo>≤</mo> <mi>q</mi> </math> are integers, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>l</mi> <mo>,</mo> <mi>q</mi> </mrow> </mfenced> <mo>=</mo> <mn>1</mn> </math> . We obtain the upper bounds in different situations.

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