Moments of the argument of automorphic L-functions for \(GL_2\)

Let f be a holomorphic Hecke eigenform of \(SL_2(\mathbb {Z})\) $S{L}_2(\mathbb{Z})$ with weight k. Denote by L(s, f) the automorphic L-function attached to f, and \(\begin{aligned} S(t, f)=\frac{1}{\pi }\arg L\Big (\frac{1}{2}+it, f\Big ), \end{aligned}\) $$\begin{array}{c}\hfill S(t,f)=\frac{1}{\pi }argL(\frac{1}{2}+it,f),\end{array}$$ where the argument is obtained by continuous variation along the straight line \(\{s\in \mathbb {C}|\Re s\ge \frac{1}{2},\, \Im s=t\}\) $\{s\in \mathbb{C}|\Re s\ge \frac{1}{2},\Im s=t\}$ , starting with the value 0 at infinity. Here, a new zero density estimate of L(s, f) in short intervals is achieved. As an application, the integral moment of S(t, f) is obtained, i.e., for \(l\in \mathbb {Z}^+\) $l\in {\mathbb{Z}}^{+}$ and sufficiently large T, \(\begin{aligned} \int _T^{T+H}|S(t, f)|^{2l}d t= \frac{(2l)!}{l!(2\pi )^{2l}}H(\log \log T)^{l}+O\big (H(\log \log T)^{l-\frac{1}{2}}\big ) \end{aligned}\) $$\begin{array}{c}\hfill {\int }_T^{T+H}{|S(t,f)|}^{2l}dt=\frac{(2l)!}{l!{(2\pi )}^{2l}}H{(loglogT)}^l+O(H{(loglogT)}^{l-\frac{1}{2}})\end{array}$$ holds for \(T^{\frac{15}{16}+\varepsilon }\le H\le T\) $T^{\frac{15}{16}+\epsilon }\le H\le T$ , which improves the previous result.