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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.17599 |
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| _version_ | 1866913586425626624 |
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| author | Çalım, Bora Iakovakis, Ioannis Long, Sophie Moffatt, Jack Wooton, Deborah |
| author_facet | Çalım, Bora Iakovakis, Ioannis Long, Sophie Moffatt, Jack Wooton, Deborah |
| contents | We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} Λ(n) Λ\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\bigl\{\lfloor m^c \rfloor \colon m \in \mathbb{N} \bigr\}$ for any non-integer $c > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Popular differences in primes along fractional powers Çalım, Bora Iakovakis, Ioannis Long, Sophie Moffatt, Jack Wooton, Deborah Number Theory 11N05 (Primary), 11P55, 11B30 (Secondary) We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} Λ(n) Λ\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$. As a combinatorial consequence, we obtain that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence $\bigl\{\lfloor m^c \rfloor \colon m \in \mathbb{N} \bigr\}$ for any non-integer $c > 2$. |
| title | Popular differences in primes along fractional powers |
| topic | Number Theory 11N05 (Primary), 11P55, 11B30 (Secondary) |
| url | https://arxiv.org/abs/2411.17599 |