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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2401.00270 |
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| _version_ | 1866916097484128256 |
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| author | Jones, Gareth A. Zvonkin, Alexander K. |
| author_facet | Jones, Gareth A. Zvonkin, Alexander K. |
| contents | Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We use theorems of Landau and Raikov to prove that the number of integers $n\le x$ with only such prime factors $q$ is asymptotic to $cx/\sqrt{\ln x}$ for a specific constant $c=0.4865\ldots$. Heuristic arguments, following Hardy and Littlewood, then yield a conjecture that the number of such primes $p\le x$ is asymptotic to $c'\int_2^x(\ln t)^{-3/2}dt$ for a constant $c'=0.8981\ldots$. The theorem, the conjecture and a similar conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann surfaces are supported by evidence from extensive computer searches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00270 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A number-theoretic problem concerning pseudo-real Riemann surfaces Jones, Gareth A. Zvonkin, Alexander K. Number Theory 11A41, 11M06, 20B25, 30F10 Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We use theorems of Landau and Raikov to prove that the number of integers $n\le x$ with only such prime factors $q$ is asymptotic to $cx/\sqrt{\ln x}$ for a specific constant $c=0.4865\ldots$. Heuristic arguments, following Hardy and Littlewood, then yield a conjecture that the number of such primes $p\le x$ is asymptotic to $c'\int_2^x(\ln t)^{-3/2}dt$ for a constant $c'=0.8981\ldots$. The theorem, the conjecture and a similar conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann surfaces are supported by evidence from extensive computer searches. |
| title | A number-theoretic problem concerning pseudo-real Riemann surfaces |
| topic | Number Theory 11A41, 11M06, 20B25, 30F10 |
| url | https://arxiv.org/abs/2401.00270 |