Prime-free discs in imaginary quadratic fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914096056631296 |
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| author | Khale, Tanmay |
| author_facet | Khale, Tanmay |
| contents | Suppose $K$ is an imaginary quadratic field, and let $N_K$ denote the field norm in the ring of integers $O_K$. Let $B(x_0,r) = \{x \in O_K: |N_K(x-x_0)| < r\}$. Let $G_K(X) = \max \{r > 0: \text{there exists } x_0 \in O_K \text{ such that } |N_K(x_0)| \leq X \text{ and } B(x_0,r) \text{ contains no primes} \}$. We show that $ G_{K}(X) \gg_K (\log X) \frac{\log_2(X) \log_4(X)}{\log_3(X)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14006 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime-free discs in imaginary quadratic fields Khale, Tanmay Number Theory Suppose $K$ is an imaginary quadratic field, and let $N_K$ denote the field norm in the ring of integers $O_K$. Let $B(x_0,r) = \{x \in O_K: |N_K(x-x_0)| < r\}$. Let $G_K(X) = \max \{r > 0: \text{there exists } x_0 \in O_K \text{ such that } |N_K(x_0)| \leq X \text{ and } B(x_0,r) \text{ contains no primes} \}$. We show that $ G_{K}(X) \gg_K (\log X) \frac{\log_2(X) \log_4(X)}{\log_3(X)}$. |
| title | Prime-free discs in imaginary quadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.14006 |