Distribution of the number of prime factors with a given multiplicity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910736798711808 |
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| author | Elma, Ertan Martin, Greg |
| author_facet | Elma, Ertan Martin, Greg |
| contents | Given an integer $k\ge2$, let $ω_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $ω_k(n)=m$ for every integer $m\ge0$. We also show that the generating function $\sum_{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod_{p} \bigl(1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum_{j=2}^\infty a_j ω_j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $Ω(n)-ω(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of the number of prime factors with a given multiplicity Elma, Ertan Martin, Greg Number Theory Given an integer $k\ge2$, let $ω_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $ω_k(n)=m$ for every integer $m\ge0$. We also show that the generating function $\sum_{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod_{p} \bigl(1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum_{j=2}^\infty a_j ω_j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $Ω(n)-ω(n)$. |
| title | Distribution of the number of prime factors with a given multiplicity |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.04574 |