Let f be a holomorphic Hecke eigenform of even weight κ for the full modular group, and let L(s, f) be the associated automorphic L-function of f. We prove that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9674_Article_IEq1.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="256" /> </InlineMediaObject> <EquationSource Format="TEX">${\int }_{T}^{2T}{\left|L\left(1/2+\text{i}t,f\right)\right|}^{2r}\text{d}t{\ll }_{f}T{\left(\text{log}T\right)}^{{r}^{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∫</mo> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </msubsup> <msup> <mrow> <mfenced close="|" open="|"> <mi>L</mi> <mfenced close=")" open="("> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mtext>i</mtext> <mi>t</mi> <mo>,</mo> <mi>f</mi> </mfenced> </mfenced> </mrow> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msup> <mtext>d</mtext> <mi>t</mi> <msub> <mo>≪</mo> <mi>f</mi> </msub> <mi>T</mi> <msup> <mrow> <mfenced close=")" open="("> <mtext>log</mtext> <mi>T</mi> </mfenced> </mrow> <msup> <mrow> <mi>r</mi> </mrow> <mn>2</mn> </msup> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10986_2025_9674_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0\le r\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> unconditionally.

Upper bounds for the moments of automorphic L-functions for GL(2)

期刊论文
发表日期：

2025年4月23日

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Minghua Chen,
Hengcai Tang

摘要

Let f be a holomorphic Hecke eigenform of even weight κ for the full modular group, and let L(s, f) be the associated automorphic L-function of f. We prove that ${\int }_{T}^{2T}{\left|L\left(1/2+\text{i}t,f\right)\right|}^{2r}\text{d}t{\ll }_{f}T{\left(\text{log}T\right)}^{{r}^{2}}$ $\int_{T}^{2 T} {|L, (1 / 2 + i t, f)|}^{2 r} d t ≪_{f} T {(log, T)}^{r^{2}}$ for $0\le r\le 1$ $0 \leq r \leq 1$ unconditionally.