Weighted Erdős-Kac Theorems via Computing Moments
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908207277932544 |
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| author | Fan, Steve |
| author_facet | Fan, Steve |
| contents | By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erdős--Kac theorem for $ω(n)$ with multiplicative weight $d_k(n)$, where $ω(n)$ denotes the number of distinct prime divisors of a positive integer $n$, and $d_k(n)$ is the $k$-fold divisor function with $k\in\mathbb{N}$. In this paper, we generalize their method to study the distribution of additive functions $f(n)$ weighted by nonnegative multiplicative functions $α(n)$ in a wide class. In particular, we establish uniform asymptotic formulas for the moments of $f(n)$ with suitable growth rates. We also prove a qualitative result on the moments which extends a theorem of Delange and Halberstam [8]. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11289 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weighted Erdős-Kac Theorems via Computing Moments Fan, Steve Number Theory 11K65, 11N60, 11N37 By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erdős--Kac theorem for $ω(n)$ with multiplicative weight $d_k(n)$, where $ω(n)$ denotes the number of distinct prime divisors of a positive integer $n$, and $d_k(n)$ is the $k$-fold divisor function with $k\in\mathbb{N}$. In this paper, we generalize their method to study the distribution of additive functions $f(n)$ weighted by nonnegative multiplicative functions $α(n)$ in a wide class. In particular, we establish uniform asymptotic formulas for the moments of $f(n)$ with suitable growth rates. We also prove a qualitative result on the moments which extends a theorem of Delange and Halberstam [8]. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem. |
| title | Weighted Erdős-Kac Theorems via Computing Moments |
| topic | Number Theory 11K65, 11N60, 11N37 |
| url | https://arxiv.org/abs/2306.11289 |