Let f(z) be a Maass cusp form and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1129_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the n-th normalized Fourier coefficient of f. In this paper, we investigate the following sums <Equation ID="Equ21"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1129_Article_Equ21.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="452" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} S_{k}(x,j):= \sum _{n \le x}\lambda _{k,f}(n^{j})= \sum _{n \le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j}) \lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>S</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mi>j</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>⋯</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> </mrow> </munder> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>n</mi> <mrow> <mn>1</mn> </mrow> <mi>j</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>n</mi> <mrow> <mn>2</mn> </mrow> <mi>j</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>⋯</mo> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>n</mi> <mrow> <mi>k</mi> </mrow> <mi>j</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for any fixed integer <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1129_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1129_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$j=1,2,3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. Our results improve the previous results in the error term.

On general divisor problems related to Maass cusp forms

期刊论文
发表日期：

2025年6月9日

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Jinzhi Feng

摘要

Let f(z) be a Maass cusp form and $\lambda _{f}(n)$ $λ_{f} (n)$ be the n-th normalized Fourier coefficient of f. In this paper, we investigate the following sums $\begin{aligned} S_{k}(x,j):= \sum _{n \le x}\lambda _{k,f}(n^{j})= \sum _{n \le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j}) \lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}), \end{aligned}$ $\begin{matrix} S_{k} (x, j) : = \sum_{n \leq x} λ_{k, f} (n^{j}) = \sum_{n \leq x} \sum_{n = n_{1} n_{2} \dots n_{k}} λ_{f} (n_{1}^{j}) λ_{f} (n_{2}^{j}) \dots λ_{f} (n_{k}^{j}), \end{matrix}$ for any fixed integer $k\ge 2$ $k \geq 2$ and $j=1,2,3,4$ $j = 1, 2, 3, 4$ . Our results improve the previous results in the error term.