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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2302.11331 |
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| _version_ | 1866912433890656256 |
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| author | Merikoski, Jori |
| author_facet | Merikoski, Jori |
| contents | We show that there exists some $δ> 0$ such that, for any set of integers $B$ with $B\cap[1,Y]\gg Y^{1-δ}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \leq X$ with $b \in B$ for any $|B|=X^{1/2-δ}$ with $δ< 1/10$ and $B \subseteq [ηX^{1/2},(1-η)X^{1/2}] \cap \mathbb{Z}$, in terms of zeros of Hecke $L$-functions on $\mathbb{Q}(i)$. We obtain the expected asymptotic formula for the number of such primes provided that the set $B$ does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if $B$ is a sparse subset of primes. For an arbitrary $B$ we obtain a lower bound for the number of primes with a weaker range for $δ$, by bounding the contribution from potential exceptional characters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11331 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Gaussian primes in sparse sets Merikoski, Jori Number Theory We show that there exists some $δ> 0$ such that, for any set of integers $B$ with $B\cap[1,Y]\gg Y^{1-δ}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \leq X$ with $b \in B$ for any $|B|=X^{1/2-δ}$ with $δ< 1/10$ and $B \subseteq [ηX^{1/2},(1-η)X^{1/2}] \cap \mathbb{Z}$, in terms of zeros of Hecke $L$-functions on $\mathbb{Q}(i)$. We obtain the expected asymptotic formula for the number of such primes provided that the set $B$ does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if $B$ is a sparse subset of primes. For an arbitrary $B$ we obtain a lower bound for the number of primes with a weaker range for $δ$, by bounding the contribution from potential exceptional characters. |
| title | On Gaussian primes in sparse sets |
| topic | Number Theory |
| url | https://arxiv.org/abs/2302.11331 |