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

Abstract

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