Let f be a Hecke eigencusp form of integral weight k and level N. Denote by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{\textrm{sym}^m f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mi>m</mi> </msup> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the nth coefficient of the mth symmetric power L-function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$L(s,\textrm{sym}^m f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <msup> <mtext>sym</mtext> <mi>m</mi> </msup> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> attached to f. In this paper, we prove that (under a certain condition about the distribution of zeros of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$L(s,\textrm{sym}^m f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <msup> <mtext>sym</mtext> <mi>m</mi> </msup> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) <Equation ID="Equ78"> <MediaObject ID="MO1"> <ImageObject Color="BlackWhite" FileRef="MediaObjects/605_2025_2117_Equ78_HTML.png" Format="PNG" Height="116" Rendition="HTML" Resolution="300" Type="Linedraw" Width="470" /> </MediaObject> </Equation>for any <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq4.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">$1\leqslant T\leqslant (\log k)^{\frac{1}{200}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>T</mi> <mo>⩽</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mn>1</mn> <mn>200</mn> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq5.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$k^{\varepsilon }\leqslant x\leqslant k^{\frac{\sqrt{\varpi }}{2}} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>k</mi> <mi>ε</mi> </msup> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <msup> <mi>k</mi> <mfrac> <msqrt> <mi>ϖ</mi> </msqrt> <mn>2</mn> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2117_Article_IEq6.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon \geqslant (\log k)^{-\frac{2}{5}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>⩾</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>.

Large sums of coefficients of symmetric power L-functions

期刊论文
发表日期：

2025年9月17日

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Jiangpeng Li,
Shu Luo

摘要

Let f be a Hecke eigencusp form of integral weight k and level N. Denote by $\lambda _{\textrm{sym}^m f}(n)$ $λ_{{sym}^{m} f} (n)$ the nth coefficient of the mth symmetric power L-function $L(s,\textrm{sym}^m f)$ $L (s, {sym}^{m} f)$ attached to f. In this paper, we prove that (under a certain condition about the distribution of zeros of $L(s,\textrm{sym}^m f)$ $L (s, {sym}^{m} f)$ ) for any $1\leqslant T\leqslant (\log k)^{\frac{1}{200}}$ $1 ⩽ T ⩽ {(log k)}^{\frac{1}{200}}$ and $k^{\varepsilon }\leqslant x\leqslant k^{\frac{\sqrt{\varpi }}{2}} $ $k^{ε} ⩽ x ⩽ k^{\frac{\sqrt{ϖ}}{2}}$ with $\varepsilon \geqslant (\log k)^{-\frac{2}{5}}$ $ε ⩾ {(log k)}^{- \frac{2}{5}}$ .