On an unconditional $\mathrm{G}{\mathrm{L}}_3$ analog of Selberg's result

Abstract

Let $F$ be a Hecke--Maass cusp form for ${\mathrm{SL}}_3(\mathbb{Z})$ with the Langlands parameter $\hat{I}{¼}_F=(\hat{I}{¼}_{F,1},\hat{I}{¼}_{F,2},\hat{I}{¼}_{F,3})$ and the associated $L$-function $L(s,F)$. Define $S_F(t)=\ddot{I}{€}^{-1}\arg L(1/2+\mathrm{i}t,F)$. When $\hat{I}{¼}_F$ 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.

22 pages, incorporates the referees' comments; to appear in SCIENCE CHINA Mathematics. Comments welcome!