The least primary factor of the multiplicative group
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911789187334144 |
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| author | Martin, Greg Nguyen, Chau |
| author_facet | Martin, Greg Nguyen, Chau |
| contents | Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of those integers $n$ for which $S(n) \ne 2$. We also give an asymptotic formula, for any prime power $q$, for the counting function of those integers $n$ for which $S(n) = q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The least primary factor of the multiplicative group Martin, Greg Nguyen, Chau Number Theory 11N25, 11N37, 11N45, 11N64, 20K01 Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of those integers $n$ for which $S(n) \ne 2$. We also give an asymptotic formula, for any prime power $q$, for the counting function of those integers $n$ for which $S(n) = q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory. |
| title | The least primary factor of the multiplicative group |
| topic | Number Theory 11N25, 11N37, 11N45, 11N64, 20K01 |
| url | https://arxiv.org/abs/2403.02548 |