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arXiv · 2502.01288

On an unconditional $\rm GL_3$ analog of Selberg's result

Qingfeng Sun·Hui Wang
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Abstract

Let $F$ be a Hecke--Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ with the Langlands parameter $\mu_{F}=\big(\mu_{F,1},\mu_{F,2},\mu_{F,3}\big)$ and the associated $L$-function $L(s, F)$. Define $S_F(t)=\pi^{-1}\arg L(1/2+\mathrm{i}t, F)$. When $\mu_{F}$ is in generic position, we establish an unconditional asymptotic formula for the moments of $S_F(t)$. Previously, such a formula was only known to hold under the Generalized Riemann Hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for $L(s, F)$, which has recently been proved by Sun and Wang in arXiv:2412.02416.

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Qingfeng Sun, Hui Wang. 2025-02-03. On an unconditional $\rm GL_3$ analog of Selberg's result. https://arxiv.org/abs/2502.01288

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