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

TL;DR

This paper proves an unconditional asymptotic formula for the moments of a phase function associated with $ m{GL}_3$ automorphic $L$-functions, extending previous results that depended on the Riemann Hypothesis.

Contribution

It establishes the first unconditional asymptotic formula for moments of $S_F(t)$ for $ m{GL}_3$ forms, using a new zero-density estimate in the spectral aspect.

Findings

Unconditional asymptotic formula for moments of $S_F(t)$

Extension of results previously conditional on GRH

Application of recent zero-density estimates

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.