Complex moments of class numbers with fundamental unit restrictions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909278329110528 |
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| author | Dousselin, Jérémy |
| author_facet | Dousselin, Jérémy |
| contents | We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit $\varepsilon_d$ is lower than $d^{1/2+α}$, where $0<α<1/2$. To do so we find an asymptotic formula for $z^{th}$-moments of such $h(d)$'s, over $d\leq x$, uniformly for a complex number $z$ in a range of the form $|z|\leq(\log x)^{1+o(1)}$, $\Re(z)\geq -1$. This is achieved by constructing a probabilistic random model for these values, which we will use to obtain estimates for the distribution function of $h(d)$ over our family. As another application, we give an asymptotic formula for the number of $d$'s such that $h(d)\leq H$ and $\varepsilon_d\leq d^{1/2+α}$ where $H$ is a large real number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex moments of class numbers with fundamental unit restrictions Dousselin, Jérémy Number Theory We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit $\varepsilon_d$ is lower than $d^{1/2+α}$, where $0<α<1/2$. To do so we find an asymptotic formula for $z^{th}$-moments of such $h(d)$'s, over $d\leq x$, uniformly for a complex number $z$ in a range of the form $|z|\leq(\log x)^{1+o(1)}$, $\Re(z)\geq -1$. This is achieved by constructing a probabilistic random model for these values, which we will use to obtain estimates for the distribution function of $h(d)$ over our family. As another application, we give an asymptotic formula for the number of $d$'s such that $h(d)\leq H$ and $\varepsilon_d\leq d^{1/2+α}$ where $H$ is a large real number. |
| title | Complex moments of class numbers with fundamental unit restrictions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.01401 |