A Prime Number Theorem for Rankin-Selberg L-functions over Number fields

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $\pi$ defined on $GL_n/E$ and $\pi'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of coprime degree where $\pi$ and $\pi'$ admit a base change lift from $\mathbb{Q}$ again given a self-contragredient assumption on at least one of the representations which lift to $\pi$ or $\pi'$.