Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$H_k(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the set of primitive holomorphic cusp forms with even integer weight <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> for the full modular group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> which is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$SL(2,\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Denote by <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{f\otimes f\otimes f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>f</mi> <mo>⊗</mo> <mi>f</mi> <mo>⊗</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the Fourier coefficients of the triple product L-function attached to f. In this article, we investigate the mean distribution of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_977_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{f\otimes f\otimes f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>f</mi> <mo>⊗</mo> <mi>f</mi> <mo>⊗</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over sparse sets in short intervals, and present an asymptotic formula for it.

Fourier coefficients of the triple product L-function over sparse sets

期刊论文
发表日期：

2025年1月3日

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Weili Yao

摘要

Let $H_k(\Gamma )$ $H_{k} (Γ)$ be the set of primitive holomorphic cusp forms with even integer weight $k\geqslant 2$ $k ⩾ 2$ for the full modular group $\Gamma $ $Γ$ which is $SL(2,\mathbb {Z})$ $S L (2, Z)$ . Denote by $\lambda _{f\otimes f\otimes f}(n)$ $λ_{f \otimes f \otimes f} (n)$ the Fourier coefficients of the triple product L-function attached to f. In this article, we investigate the mean distribution of $\lambda _{f\otimes f\otimes f}(n)$ $λ_{f \otimes f \otimes f} (n)$ over sparse sets in short intervals, and present an asymptotic formula for it.