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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2605.09807 |
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| _version_ | 1866909031014072320 |
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| author | Huang, Tinghao Zhao, Shifan |
| author_facet | Huang, Tinghao Zhao, Shifan |
| contents | For a primitive Hecke-Maass cusp form $ϕ$ of level $N$ with the $n$-th Hecke eigenvalue $λ_ϕ(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: \[
|λ_ϕ(p)| \leq 2. \] 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 $\{ϕ_i\}$, 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 $ϕ_i$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09807 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Ramanujan Primes for Hecke-Maass Cusp Forms Huang, Tinghao Zhao, Shifan Number Theory For a primitive Hecke-Maass cusp form $ϕ$ of level $N$ with the $n$-th Hecke eigenvalue $λ_ϕ(n)$ and a prime number $p\nmid N$, the celebrated Ramanujan conjecture at $p$ asserts the following sharp upper bound: \[ |λ_ϕ(p)| \leq 2. \] 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 $\{ϕ_i\}$, 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 $ϕ_i$'s. |
| title | On Ramanujan Primes for Hecke-Maass Cusp Forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.09807 |