Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {A}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(1,n)\text {-th}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext>-th</mtext> </mrow> </math></EquationSource> </InlineEquation> Fourier coefficient of a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\text {SL}(3, \mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Hecke eigenform or the ternary divisor function <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d_3(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Let Q(x, y) be a symmetric positive definite quadratic form. This article establishes an asymptotic formula with a power-saving error term for the following sum <Equation ID="Equ64"> <EquationSource Format="TEX">$\begin{aligned} \sum _{1 \leqslant m \leqslant X} \sum _{1 \leqslant n\leqslant Y} \mathcal {A}(Q(m,n)), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>m</mi> <mo>⩽</mo> <mi>X</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>n</mi> <mo>⩽</mo> <mi>Y</mi> </mrow> </munder> <mi mathvariant="script">A</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$X>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$Y\leqslant X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Y</mi> <mo>⩽</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>.

Sum of the $\text {GL}(3)$ Fourier coefficients over quadratics

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发表日期：

2026年4月13日

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Himanshi Chanana,
Saurabh Kumar Singh

摘要

Let $\mathcal {A}(n)$ $A (n)$ denote the $(1,n)\text {-th}$ $(1, n) -th$ Fourier coefficient of a $\text {SL}(3, \mathbb {Z})$ $SL (3, Z)$ Hecke eigenform or the ternary divisor function $d_3(n)$ $d_{3} (n)$ . Let Q(x, y) be a symmetric positive definite quadratic form. This article establishes an asymptotic formula with a power-saving error term for the following sum $\begin{aligned} \sum _{1 \leqslant m \leqslant X} \sum _{1 \leqslant n\leqslant Y} \mathcal {A}(Q(m,n)), \end{aligned}$ $\begin{matrix} \sum_{1 ⩽ m ⩽ X} \sum_{1 ⩽ n ⩽ Y} A (Q (m, n)), \end{matrix}$ where $X>1$ $X > 1$ and $Y\leqslant X$ $Y ⩽ X$ .

Sum of the \(\text {GL}(3)\) Fourier coefficients over quadratics

摘要