On Ramanujan Primes for Hecke-Maass Cusp Forms

For a primitive Hecke-Maass cusp form $\phi$ of level $N$ with the $n$-th Hecke eigenvalue ${\lambda }_{\phi }(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: $$\mathrm{\mid }{\lambda }_{\phi }(p)\mathrm{\mid }\le 2.$$ In this work, we determine an upper bound for the least prime $p$ at which the Ramanujan conjecture holds for two or three distinct primitive Hecke-Maass cusp forms simultaneously. Moreover, given a set of distinct primitive Hecke-Maass cusp forms $\{{\phi }_i\}$, we also provide a lower bound for the lower natural density of the set of primes at which the Ramanujan conjecture holds for at least one of the ${\phi }_i$’s.