On the number of prime factors with a given multiplicity over h-free and h-full numbers
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912031836209152 |
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| author | Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru |
| author_facet | Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru |
| contents | Let $k$ and $n$ be natural numbers. Let $ω_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $ω_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $ω_1(n)$ has normal order $\log \log n$ over $h$-free numbers, $ω_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions $ω_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $ω_k(n)$ with $k > h$ do not have normal order over $h$-full numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of prime factors with a given multiplicity over h-free and h-full numbers Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru Number Theory 11N05, 11N37 Let $k$ and $n$ be natural numbers. Let $ω_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $ω_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $ω_1(n)$ has normal order $\log \log n$ over $h$-free numbers, $ω_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions $ω_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $ω_k(n)$ with $k > h$ do not have normal order over $h$-full numbers. |
| title | On the number of prime factors with a given multiplicity over h-free and h-full numbers |
| topic | Number Theory 11N05, 11N37 |
| url | https://arxiv.org/abs/2409.11275 |