Prime number theorems for Rankin-Selberg L-functions over number fields

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal representations of GLm(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{A} $$\end{document}E) × GLm′ (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{A} $$\end{document}F) over cyclic algebraic number fields E and F which are invariant under the Galois action, by exploiting a result proved by Arthur and Clozel, and prove a prime number theorem for this L-function.