On an unconditional spectral analog of Selberg's result on $S(t)$

TL;DR

This paper derives an asymptotic formula for the moments of the argument function associated with Hecke--Maass cusp forms, extending Selberg's results without relying on the Generalized Riemann Hypothesis.

Contribution

It provides the first unconditional asymptotic formula for the moments of $S_j(t)$ related to Hecke--Maass forms, generalizing classical results.

Findings

Asymptotic formula for moments of $S_j(t)$ established

Results hold without assuming GRH

Advances understanding of spectral analogs of Selberg's results

Abstract

Let $S_j(t)=\frac{1}{\pi}\arg L(1/2+it, u_j)$, where $u_j$ is an even Hecke--Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplacian eigenvalue $\lambda_j=\frac{1}{4}+t_j^2$. Without assuming the GRH, we establish an asymptotic formula for the moments of $S_j(t)$.