Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\lambda _f(n)\}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> denote the normalized Hecke eigenvalues of a holomorphic cusp form f, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda _{f,j}=\lambda _f\cdots \lambda _f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow /> <mo>∗</mo> <mo>⋯</mo> <mrow /> <mo>∗</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be the j-fold convolution of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda _f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we demonstrate that, for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$j\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X^{\frac{2}{3}+\varepsilon }<H<X^{1-\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mrow> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo><</mo> <mi>H</mi> <mo><</mo> <msup> <mi>X</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>ε</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, the following inequality holds: <Equation ID="Equ3"> <EquationSource Format="TEX">$\begin{aligned} \sum _{X \le n \le 2X}\lambda _{f,j}(n)\lambda _{f}^2(n+h)\ll _{f,A,\varepsilon }X(\log X)^{-A} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mi>X</mi> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>2</mn> <mi>X</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>λ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>≪</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>ε</mi> </mrow> </msub> <mi>X</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mi>A</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all but <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$O_{f,A,\varepsilon }(H(\log X)^{-3A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>O</mi> <mrow> <mi>f</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>ε</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>3</mn> <mi>A</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> integers <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h\in [1,H]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>H</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Additionally, when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1\le j\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, more favorable results can be achieved.

Correlations of the j-fold convolution of Hecke eigenvalues and Hecke eigenvalue squares

期刊论文
发表日期：

2025年10月11日

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Chengchao Huang

摘要

Let $\{\lambda _f(n)\}_{n\ge 1}$ ${λ_{f} (n)}_{n \geq 1}$ denote the normalized Hecke eigenvalues of a holomorphic cusp form f, and let $\lambda _{f,j}=\lambda _f*\cdots *\lambda _f$ $λ_{f, j} = λ_{f} * \dots * λ_{f}$ be the j-fold convolution of $\lambda _f$ $λ_{f}$ . In this paper, we demonstrate that, for $j\ge 3$ $j \geq 3$ and $X^{\frac{2}{3}+\varepsilon }<H<X^{1-\varepsilon }$ $X^{\frac{2}{3} + ε} < H < X^{1 - ε}$ , as $X\rightarrow \infty $ $X \to \infty$ , the following inequality holds: $\begin{aligned} \sum _{X \le n \le 2X}\lambda _{f,j}(n)\lambda _{f}^2(n+h)\ll _{f,A,\varepsilon }X(\log X)^{-A} \end{aligned}$ $\begin{matrix} \sum_{X \leq n \leq 2 X} λ_{f, j} (n) λ_{f}^{2} (n + h) ≪_{f, A, ε} X {(log X)}^{- A} \end{matrix}$ for all but $O_{f,A,\varepsilon }(H(\log X)^{-3A})$ $O_{f, A, ε} (H {(log X)}^{- 3 A})$ integers $h\in [1,H]$ $h \in [1, H]$ . Additionally, when $1\le j\le 2$ $1 \leq j \leq 2$ , more favorable results can be achieved.