Distribution of $ω(n)$ over $h$-free and $h$-full numbers
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917777079533568 |
|---|---|
| author | Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru |
| author_facet | Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru |
| contents | Let $ω(n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $ω(n)$ has normal order $\log \log n$ over naturals. In this work, we establish the first and the second moments of $ω(n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $ω(n)$ has normal order $\log \log n$ over these subsets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of $ω(n)$ over $h$-free and $h$-full numbers Das, Sourabhashis Kuo, Wentang Liu, Yu-Ru Number Theory 11N37, 11N05, 11N56 Let $ω(n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $ω(n)$ has normal order $\log \log n$ over naturals. In this work, we establish the first and the second moments of $ω(n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $ω(n)$ has normal order $\log \log n$ over these subsets. |
| title | Distribution of $ω(n)$ over $h$-free and $h$-full numbers |
| topic | Number Theory 11N37, 11N05, 11N56 |
| url | https://arxiv.org/abs/2409.10430 |