The Ramanujan conjecture for cuspidal automorphic representations of general linear groups

Let $k$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_k)$. A local component $\pi_v$ is tempered when its local representation is tempered. Ramanujan conjecture for $\operatorname{GL}_n$. The representation $\pi_v$ is tempered for every place $v$. This is the standard generalized Ramanujan assertion for cuspidal automorphic representations of general linear groups and is used in the paper as an assumption for some conditional consequences.