On Ramanujan Primes for Hecke-Maass Cusp Forms

TL;DR

This paper establishes bounds on the smallest primes where Ramanujan's conjecture holds for multiple Hecke-Maass cusp forms and analyzes the density of primes satisfying the conjecture for at least one form.

Contribution

It provides new bounds for the least prime where Ramanujan's conjecture is true for multiple forms and estimates the density of such primes across forms.

Findings

Upper bounds for the least prime satisfying Ramanujan for multiple forms

Lower bounds on the density of primes where Ramanujan holds for at least one form

Results apply to sets of two or three primitive Hecke-Maass cusp forms

Abstract

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: \[ |\lambda_{\phi}(p)| \leq 2. \] 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.