Moments of automorphic L-functions in the strip $\frac{1}{2}<\sigma <1$

Abstract

Let $\pi$ be a self-contragredient irreducible unitary cuspidal representation of $GL_m({\mathbb {Q}}_{{\mathbb {A}}})$ with $m\ge 2$, and $L(s,\pi )$ be the automorphic L-function attached to $\pi$. Under the Generalized Riemann Hypothesis for $L(s, \pi )$, the asymptotic formula

$$\begin{aligned} \int _1^{T}\left| L\left( \sigma +it, \pi \right) \right| ^{2k}d t =T\sum _{n=1}^\infty \frac{\nu _\pi (n, k)^2}{n^{2\sigma }} + O\left( T^{1-\frac{2\sigma -1}{2(3-2\sigma )}+\varepsilon }\right) \end{aligned}$$

is obtained for $\frac{1}{2}<\sigma <1,\,k\in {\mathbb {Z}}^+$ as $T\rightarrow \infty$, where $\varepsilon >0$ is arbitrarily small and O depends on $\pi , k, \sigma$, and $\varepsilon$. Here, $\nu _\pi (n, k)$ is the coefficient of $L^k(s, \pi )$.