A $\mathrm{GL}_3$ level analog of Selberg's result for $S(t)$

Abstract

Let $\pi$ be a spherical cuspidal $\mathrm{GL}_3$ newform of prime level $N$ and trivial central character, and define $S(t,\pi)=\pi^{-1}\arg L(1/2+\textrm{i}t,\pi)$. Assuming the generalized Riemann hypothesis for $L(s,\pi)$, we establish an asymptotic formula for each fixed weighted moment of $S(t,\pi)$ as $N\to\infty$, for fixed $t>0$. Consequently, under the same weights, $S(t,\pi)/\sqrt{\log\log N}$ converges in distribution to a centered Gaussian distribution of variance $1/(2\pi^2)$. The key input is a weighted orthogonality formula derived from the Kuznetsov formula of prime level.