<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_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 <i>n</i>th normalized Fourier coefficient of the primitive holomorphic Hecke eigenforms of even integral weight <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_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> for the full modular group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_Article_IEq3.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>. In this paper, we investigate the asymptotic formula of the power sum <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_Article_IEq4.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{n \le x} \lambda ^{l}_{f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mi>f</mi> <mi>l</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$l=2m \ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_Article_IEq6.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="149" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{a^{2}+b^{2}\le x} \lambda ^{l}_{f} (a^{2}+b^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mi>f</mi> <mi>l</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1000_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$l=2m $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, and improve on previous error estimates.</p>

Asymptotic formula of Fourier coefficients of cusp forms over sum of two squares

期刊论文
发表日期：

2025年1月31日

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Jinzhi Feng

摘要

Let $\lambda _{f}(n)$ $λ_{f} (n)$ be the nth normalized Fourier coefficient of the primitive holomorphic Hecke eigenforms of even integral weight $k \ge 2$ $k \geq 2$ for the full modular group $SL(2,\mathbb {Z})$ $S L (2, Z)$ . In this paper, we investigate the asymptotic formula of the power sum $\sum _{n \le x} \lambda ^{l}_{f}(n)$ $\sum_{n \leq x} λ_{f}^{l} (n)$ for $l=2m \ge 6$ $l = 2 m \geq 6$ and $\sum _{a^{2}+b^{2}\le x} \lambda ^{l}_{f} (a^{2}+b^{2})$ $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{l} (a^{2} + b^{2})$ for $l=2m $ $l = 2 m$ , and improve on previous error estimates.