<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$f$</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> be a Hecke-Maass cusp form for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm{SL}_2(\mathbb{Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">SL</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with normalized Fourier coefficients <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq3.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> and Laplace eigenvalue <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$1/4+\mu_f^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <msubsup> <mi>μ</mi> <mi>f</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$g$</EquationSource> <EquationSource Format="MATHML"><math> <mi>g</mi> </math></EquationSource> </InlineEquation> be a Hecke-Maass cusp form for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm{SL}_2(\mathbb{Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">SL</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with normalized Fourier coefficients <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_g(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>g</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we study the asymptotic of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum_{n \leq X}\lambda_{1\boxplus(f\times g)}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>X</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow> <mn>1</mn> <mo>⊞</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>×</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and get the explicit dependence of the error term on the spectral parameter <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1544_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Sums of Fourier coefficients of certain Eisenstein series of GL(5)

期刊论文
发表日期：

2025年6月20日

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Ch. Shao,
H. Zhang

摘要

Let $f$ $f$ be a Hecke-Maass cusp form for $\mathrm{SL}_2(\mathbb{Z})$ ${SL}_{2} (Z)$ with normalized Fourier coefficients $\lambda_f(n)$ $λ_{f} (n)$ and Laplace eigenvalue $1/4+\mu_f^2$ $1 / 4 + μ_{f}^{2}$ . Let $g$ $g$ be a Hecke-Maass cusp form for $\mathrm{SL}_2(\mathbb{Z})$ ${SL}_{2} (Z)$ with normalized Fourier coefficients $\lambda_g(n)$ $λ_{g} (n)$ . In this paper, we study the asymptotic of $\sum_{n \leq X}\lambda_{1\boxplus(f\times g)}(n)$ $\sum_{n \leq X} λ_{1 ⊞ (f \times g)} (n)$ and get the explicit dependence of the error term on the spectral parameter $\mu_f$ $μ_{f}$ .