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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2504.11452 |
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| _version_ | 1866910912001081344 |
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| author | Martin, Greg Simpson, Reginald M. |
| author_facet | Martin, Greg Simpson, Reginald M. |
| contents | The structure of the multiplicative group $M_n = ({\mathbb Z}/n{\mathbb Z})^\times$ encodes a great deal of arithmetic information about the integer $n$ (examples include $ϕ(n)$, the Carmichael function $λ(n)$, and the number $ω(n)$ of distinct prime factors of $n$). We examine the invariant factor structure of $M_n$ for typical integers $n$, that is, the decomposition $M_n \cong {\mathbb Z}/d_1{\mathbb Z} \times {\mathbb Z}/d_2{\mathbb Z} \times \cdots \times {\mathbb Z}/d_k{\mathbb Z}$ where $d_1\mid d_2\mid\cdots\mid d_k$. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically $1/2$ of the invariant factors equal ${\mathbb Z}/2{\mathbb Z}$, asymptotically $1/4$ of them equal ${\mathbb Z}/12{\mathbb Z}$, asymptotically $1/12$ of them equal ${\mathbb Z}/120{\mathbb Z}$, and so on. Furthermore, for positive integers $k$, we establish a theorem of Erdős-Kac type for the number of invariant factors of $M_n$ that equal ${\mathbb Z}/k{\mathbb Z}$, except that the distribution is not a normal distribution but rather a skew-normal or related distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The universal profile of the invariant factors of $({\mathbb Z}/n{\mathbb Z})^\times$ Martin, Greg Simpson, Reginald M. Number Theory The structure of the multiplicative group $M_n = ({\mathbb Z}/n{\mathbb Z})^\times$ encodes a great deal of arithmetic information about the integer $n$ (examples include $ϕ(n)$, the Carmichael function $λ(n)$, and the number $ω(n)$ of distinct prime factors of $n$). We examine the invariant factor structure of $M_n$ for typical integers $n$, that is, the decomposition $M_n \cong {\mathbb Z}/d_1{\mathbb Z} \times {\mathbb Z}/d_2{\mathbb Z} \times \cdots \times {\mathbb Z}/d_k{\mathbb Z}$ where $d_1\mid d_2\mid\cdots\mid d_k$. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically $1/2$ of the invariant factors equal ${\mathbb Z}/2{\mathbb Z}$, asymptotically $1/4$ of them equal ${\mathbb Z}/12{\mathbb Z}$, asymptotically $1/12$ of them equal ${\mathbb Z}/120{\mathbb Z}$, and so on. Furthermore, for positive integers $k$, we establish a theorem of Erdős-Kac type for the number of invariant factors of $M_n$ that equal ${\mathbb Z}/k{\mathbb Z}$, except that the distribution is not a normal distribution but rather a skew-normal or related distribution. |
| title | The universal profile of the invariant factors of $({\mathbb Z}/n{\mathbb Z})^\times$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.11452 |