Prime values of Ramanujan's tau function
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915592612610048 |
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| author | Xiong, Boyuan |
| author_facet | Xiong, Boyuan |
| contents | We study the prime values of Ramanujan's tau function $τ(n)$. Lehmer found that $n=251^2=63001$ is the smallest $n$ such that $τ(n)$ is prime: $$τ(251^2)=-80561663527802406257321747.$$ We prove that in most arithmetic progressions (mod 23), the prime values $τ$ belonging to the progression form a thin set. As a consequence, there exists a set of primes of Dirichlet density $\frac{9}{11}$ which are not values of $τ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Prime values of Ramanujan's tau function Xiong, Boyuan Number Theory We study the prime values of Ramanujan's tau function $τ(n)$. Lehmer found that $n=251^2=63001$ is the smallest $n$ such that $τ(n)$ is prime: $$τ(251^2)=-80561663527802406257321747.$$ We prove that in most arithmetic progressions (mod 23), the prime values $τ$ belonging to the progression form a thin set. As a consequence, there exists a set of primes of Dirichlet density $\frac{9}{11}$ which are not values of $τ$. |
| title | Prime values of Ramanujan's tau function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.12073 |