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.
Categories: math.NT