<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_{\pi }(n,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the (<i>n</i>, 1)-th Fourier coefficient of the Hecke-Maass cusp form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathrm SL_3(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">S</mi> <msub> <mi>L</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \omega (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum <Equation ID="Equ56"> <EquationSource Format="TEX">$ \sum _{\begin{array}{c} n_1,\cdots ,n_\ell ,n_{\ell +1}\in \mathbb {Z}_+ \\ n=n_1^r+\cdots +n_{\ell }^r+n_{\ell +1}^s \end{array}} A_{\pi }(n,1)\omega \left( n/X\right) , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mo>∑</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>n</mi> <mi>ℓ</mi> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>n</mi> <mo>=</mo> <msubsup> <mi>n</mi> <mn>1</mn> <mi>r</mi> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>n</mi> <mrow> <mi>ℓ</mi> </mrow> <mi>r</mi> </msubsup> <mo>+</mo> <msubsup> <mi>n</mi> <mrow> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>s</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mrow> </munder> <msub> <mi>A</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>ω</mi> <mfenced close=")" open="("> <mi>n</mi> <mo stretchy="false">/</mo> <mi>X</mi> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$r\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$s\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\ell \ge 2^{r-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <msup> <mn>2</mn> <mrow> <mi>r</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> are integers.</p>

On $\mathrm GL_3$ Fourier coefficients over values of mixed powers

期刊论文
发表日期：

2026年6月17日

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Qingfeng Sun,
Yanxue Yu

摘要

Let $A_{\pi }(n,1)$ $A_{π} (n, 1)$ be the (n, 1)-th Fourier coefficient of the Hecke-Maass cusp form $\pi $ $π$ for $\mathrm SL_3(\mathbb {Z})$ $S L_{3} (Z)$ and $ \omega (x)$ $ω (x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $ \sum _{\begin{array}{c} n_1,\cdots ,n_\ell ,n_{\ell +1}\in \mathbb {Z}_+ \\ n=n_1^r+\cdots +n_{\ell }^r+n_{\ell +1}^s \end{array}} A_{\pi }(n,1)\omega \left( n/X\right) , $ $\sum_{\begin{matrix} n_{1}, \dots, n_{ℓ}, n_{ℓ + 1} \in Z_{+} \\ n = n_{1}^{r} + \dots + n_{ℓ}^{r} + n_{ℓ + 1}^{s} \end{matrix}} A_{π} (n, 1) ω (n / X),$ where $r\ge 2$ $r \geq 2$ , $s\ge 2$ $s \geq 2$ and $\ell \ge 2^{r-1}$ $ℓ \geq 2^{r - 1}$ are integers.

On \(\mathrm GL_3\) Fourier coefficients over values of mixed powers

摘要