Covering integers by $x^2 + dy^2$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913567192645632 |
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| author | Green, Ben Soundararajan, Kannan |
| author_facet | Green, Ben Soundararajan, Kannan |
| contents | What proportion of integers $n \leqslant N$ may be expressed as $x^2 + dy^2$ for some $d \leqslant Δ$, with $x,y $ integers? Writing $Δ$ as $(\log N)^{\log 2} 2^{α\sqrt{\log \log N}}$ for some $α\in (-\infty, \infty)$, we show that the answer is $Φ(α) + o(1)$, where $Φ$ is the Gaussian distribution function $Φ(α) = \frac{1}{2π} \int^α_{-\infty} e^{-x^2/2} dx$.
A consequence of this is a phase transition: almost none of the integers $n \leqslant N$ can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 - \varepsilon}$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 + \varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04817 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covering integers by $x^2 + dy^2$ Green, Ben Soundararajan, Kannan Number Theory What proportion of integers $n \leqslant N$ may be expressed as $x^2 + dy^2$ for some $d \leqslant Δ$, with $x,y $ integers? Writing $Δ$ as $(\log N)^{\log 2} 2^{α\sqrt{\log \log N}}$ for some $α\in (-\infty, \infty)$, we show that the answer is $Φ(α) + o(1)$, where $Φ$ is the Gaussian distribution function $Φ(α) = \frac{1}{2π} \int^α_{-\infty} e^{-x^2/2} dx$. A consequence of this is a phase transition: almost none of the integers $n \leqslant N$ can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 - \varepsilon}$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 + \varepsilon}$. |
| title | Covering integers by $x^2 + dy^2$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.04817 |