The Piatetski-Shapiro prime number theorem
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918235029372928 |
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| author | Guo, Lingyu guo, Victor Zhenyu Lu, Li |
| author_facet | Guo, Lingyu guo, Victor Zhenyu Lu, Li |
| contents | The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro sequence for $c \in (1,2)$. In this article, we prove there are infinitely many Piatetski-Shapiro prime numbers for $1 < c < 1.1612\dots$ with an asymptotic formula. As a key idea, we prove a new bound for related type $I$ sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10391 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Piatetski-Shapiro prime number theorem Guo, Lingyu guo, Victor Zhenyu Lu, Li Number Theory 11B83, 11L07, 11N05 The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro sequence for $c \in (1,2)$. In this article, we prove there are infinitely many Piatetski-Shapiro prime numbers for $1 < c < 1.1612\dots$ with an asymptotic formula. As a key idea, we prove a new bound for related type $I$ sum. |
| title | The Piatetski-Shapiro prime number theorem |
| topic | Number Theory 11B83, 11L07, 11N05 |
| url | https://arxiv.org/abs/2505.10391 |