On a level analog of Selberg's result on $S(t)$

Abstract

Let $S(t,f)=\pi^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect analogue of Selberg's classical work on $S(t)$. As a consequence, we derive a weighted central limit theorem for the distribution of $S(t,f)$ normalized by $\sqrt{\log\log q}$. To this end, we develop a precise approximation for $S(t,f)$ via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of $L$-functions.