On the asymptotics of the shifted sums of Hecke eigenvalue squares

Abstract

Abstract The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>X</m:mi> <m:mrow> <m:mfrac> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mi>ϵ</m:mi> </m:mrow> </m:msup> <m:mo>&lt;</m:mo> <m:mi>H</m:mi> <m:mo>&lt;</m:mo> <m:msup> <m:mi>X</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>ϵ</m:mi> </m:mrow> </m:msup> </m:mrow> </m:math> {X^{\frac{2}{3}+\epsilon}&lt;H&lt;X^{1-\epsilon}} there are constants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>h</m:mi> </m:msub> </m:math> {B_{h}} such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:munder> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> </m:munder> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>f</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:msub> <m:mi>λ</m:mi> <m:mi>f</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mi>h</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>O</m:mi> <m:mrow> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>ϵ</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mi>X</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mi>A</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \sum_{X\leq n\leq 2X}\lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,% \epsilon}(X(\log X)^{-A}) for all but <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>O</m:mi> <m:mrow> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>ϵ</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mi>X</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mi>A</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {O_{f,A,\epsilon}(H(\log X)^{-3A})} integers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>H</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {h\in[1,H]} where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>f</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {\{\lambda_{f}(n)\}_{n\geq 1}} are normalized Hecke eigenvalues of a fixed holomorphic cusp form f . Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {m_{1}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {m_{2}} . In order to treat <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {m_{2}} , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {m_{1}} , we apply Parseval’s identity and Gallagher’s lemma.

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