I heard that the Langlands functoriality conjecture implies the Ramanujan conjecture for $GL(2)$. especially for the Maass form.


There are various versions of the Langlands functoriality concerning to which groups are associated.


I am wondering which version of the Langlands functorial conjecture could prove the Ramanujan conjecture for $GL(n)$ completely?



Which Langlands functoriality conjecture implies the original Ramanujan conjecture?
Let $F$ be a number field, let $\mathbb{A}\_F$ be the ring of adeles of $F$, and let $\mathcal{A}(n)$ be the set of cuspidal automorphic representation of $\mathrm{GL}\_n(\mathbb{A}\_F)$ with unitary central character. For $\pi\in\mathcal{A}(n)$, I will express the generalized Ramanujan conjecture (GRC) for $\pi$ as the conjectural bound


$$|\lambda\_{\pi}(\mathfrak{a})|\ll\_{n,\epsilon}\mathrm{N}\mathfrak{a}^{\epsilon}$$


for all $\epsilon>0$, where $\lambda\_{\pi}(\mathfrak{a})$ is the Hecke eigenvalue at $\mathfrak{a}$. One way to approach GRC is to study the moments


$$\sum\_{\mathrm{N}\mathfrak{a}\leq x}|\lambda\_{\pi}(\mathfrak{a})|^{2k},$$


where $k\geq 1$ is a natural number. When $k=1$, this is bounded from above by


$$\sum\_{\mathrm{N}\mathfrak{a}\leq x}\lambda\_{\pi\times\widetilde{\pi}}(\mathfrak{a}),$$


where $\pi\times\pi'$ denotes the Rankin-Selberg convolution. The work of Jacquet, Piatetski-Shapiro, and Shalika establishes the basic properties of the Rankin-Selberg $L$-function $L(s,\pi\times\widetilde{\pi})$. These properties (particularly the absolute convergence of the Euler product that defines $L(s,\pi\times\widetilde{\pi})$ in the region $\mathrm{Re}(s)>1$) imply the first nontrivial bound: $|\lambda\_{\pi}(\mathfrak{a})|\ll\_{n,\epsilon}\mathrm{N}\mathfrak{a}^{1/2+\epsilon}$.


To study $k=4$, the pertinent Dirichlet series (which is conjecturally an $L$-function with an analytic continuation and functional equation) is $L(s,\pi\times\widetilde{\pi}\times\pi\times\widetilde{\pi})$. For $k=6$, we need $L(s,\pi\times\widetilde{\pi}\times\pi\times\widetilde{\pi}\times\pi\times\widetilde{\pi})$. Hopefully the pattern is clear. We expect each Dirichlet series in this sequence converges absolutely for $\mathrm{Re}(s)>1$. That would suffice to prove GRC (take $k$ to be sufficiently large in terms of $\epsilon$).


Such a region of absolute convergence follows immediately if all of these Dirichlet series are in fact products of $L$-functions of cuspidal automorphic representations (as above). Equivalently, each of these higher-order convolutions are isobaric sums of cuspidal automorphic representations. One way to express the Langlands functoriality conjecture for $\mathrm{GL}\_n$, at least as it pertains to the question, is that if $\pi\in\mathcal{A}(n)$ and $\pi'\in\mathcal{A}(n')$, then there exists an isobaric sum of cuspidal automorphic representations, say $\pi\boxtimes\pi'$, such that


$$L(s,\pi\times\pi')=L(s,\pi\boxtimes\pi').$$


