Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_999_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{\textrm{sym}^2 f} (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the nth coefficient of the symmetric square L-function attached to a holomorphic cusp form f for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_999_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_2({{\mathbb {Z}}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we study the asymptotic behaviour of <Equation ID="Equ56"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_999_Article_Equ56.gif" Format="GIF" Height="42" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{n\leqslant x} \lambda _{\textrm{sym}^2 f} (n^m), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and establish asymptotic formulae for <Equation ID="Equ57"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_999_Article_Equ57.gif" Format="GIF" Height="42" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{n\leqslant x}|\lambda _{\textrm{sym}^2 f} (n^m)|. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </munder> <mrow> <mo stretchy="false">|</mo> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Moreover, we prove an equidistribution result on the signs of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_999_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{\textrm{sym}^2 f} (p^m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <msup> <mtext>sym</mtext> <mn>2</mn> </msup> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We are able to refine or extend previous results.

The distribution of coefficients of symmetric square L-functions over sparse sequences

期刊论文
发表日期：

2025年2月3日

查看全文

Shu Luo,
Huixue Lao

摘要

Let $\lambda _{\textrm{sym}^2 f} (n)$ $λ_{{sym}^{2} f} (n)$ be the nth coefficient of the symmetric square L-function attached to a holomorphic cusp form f for $\textrm{SL}_2({{\mathbb {Z}}})$ ${SL}_{2} (Z)$ . In this paper, we study the asymptotic behaviour of $\begin{aligned} \sum _{n\leqslant x} \lambda _{\textrm{sym}^2 f} (n^m), \end{aligned}$ $\begin{matrix} \sum_{n ⩽ x} λ_{{sym}^{2} f} (n^{m}), \end{matrix}$ and establish asymptotic formulae for $\begin{aligned} \sum _{n\leqslant x}|\lambda _{\textrm{sym}^2 f} (n^m)|. \end{aligned}$ $\begin{matrix} \sum_{n ⩽ x} | λ_{{sym}^{2} f} (n^{m}) | . \end{matrix}$ Moreover, we prove an equidistribution result on the signs of $\lambda _{\textrm{sym}^2 f} (p^m)$ $λ_{{sym}^{2} f} (p^{m})$ . We are able to refine or extend previous results.