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arXiv · 2605.09807

On Ramanujan Primes for Hecke-Maass Cusp Forms

Tinghao Huang·Shifan Zhao
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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.

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Tinghao Huang, Shifan Zhao. 2026-05-10. On Ramanujan Primes for Hecke-Maass Cusp Forms. https://arxiv.org/abs/2605.09807

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