On an unconditional $\rm GL_3$ analog of Selberg's result
Qingfeng Sun, Hui Wang
TL;DR
We address unconditional asymptotics for the moments of $S_F(t)=\pi^{-1}\arg L(\tfrac{1}{2}+i t, F)$ for a GL$_3$ Hecke--Maass cusp form $F$ with generic Langlands parameter. The method combines a GL$_3$ Kuznetsov-type summation formula, GL$_3$ Hecke relations, and a recently established weighted zero-density estimate to approximate $S_F(t)$ by a short Dirichlet polynomial plus a controllable remainder. The main results show that the even moments of $S_F(t)$ grow as explicit powers of $\log\log T$, odd moments vanish, and the normalized $S_F(t)/\sqrt{\log\log T}$ converges to a Gaussian with variance $(2\pi^2)^{-1}$; in particular, this yields an unconditional Selberg-type moment formula for GL$_3$ and a Gaussian limiting distribution. These findings extend previous GRH-dependent results and point toward unconditional analogues for GL$_n$ in future work.
Abstract
Let $F$ be a Hecke--Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ with the Langlands parameter $μ_{F}=\big(μ_{F,1},μ_{F,2},μ_{F,3}\big)$ and the associated $L$-function $L(s, F)$. Define $S_F(t)=π^{-1}\arg L(1/2+\mathrm{i}t, F)$. When $μ_{F}$ 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.