<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1046_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="215" /> </InlineMediaObject> <EquationSource Format="TEX">$S_j(t)=\pi ^{-1}\arg L(1/2+it, u_j)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>π</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>arg</mo> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> <mo>,</mo> <msub> <mi>u</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1046_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$u_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> is an even Hecke–Maass cusp form for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1046_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm SL_2(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">S</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with Laplacian eigenvalue <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1046_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _j=1/4+t_j^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we establish an unconditional asymptotic formula for the moments of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1046_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$S_j(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On an unconditional spectral analog of Selberg’s result on S(t)

期刊论文
发表日期：

2025年3月4日

查看全文

Qingfeng Sun,
Hui Wang

摘要

Let $S_j(t)=\pi ^{-1}\arg L(1/2+it, u_j)$ $S_{j} (t) = π^{- 1} arg L (1 / 2 + i t, u_{j})$ , where $u_j$ $u_{j}$ is an even Hecke–Maass cusp form for $\mathrm SL_2(\mathbb {Z})$ $S L_{2} (Z)$ with Laplacian eigenvalue $\lambda _j=1/4+t_j^2$ $λ_{j} = 1 / 4 + t_{j}^{2}$ . In this paper, we establish an unconditional asymptotic formula for the moments of $S_j(t)$ $S_{j} (t)$ .