A statistical investigation of a divisor-sum function
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914450929352704 |
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| author | Aidun, Ivan Thompson, Lola |
| author_facet | Aidun, Ivan Thompson, Lola |
| contents | The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the practical numbers. We prove that $S_s(n)/n$ has a continuous asymptotic distribution function, and that its values are dense in the interval $[0,\infty)$. We also establish mean value computations for $S_s(n)$ and $S_s(n)/n$, and provide uniform bounds for the higher order moments of $S_s(n)/n$. The main novelty in this paper is that we highlight a new method of Lebowitz-Lockard and Pollack that is useful for showing that certain functions have a continuous distribution function where classical methods sometimes fail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05284 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A statistical investigation of a divisor-sum function Aidun, Ivan Thompson, Lola Number Theory The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the practical numbers. We prove that $S_s(n)/n$ has a continuous asymptotic distribution function, and that its values are dense in the interval $[0,\infty)$. We also establish mean value computations for $S_s(n)$ and $S_s(n)/n$, and provide uniform bounds for the higher order moments of $S_s(n)/n$. The main novelty in this paper is that we highlight a new method of Lebowitz-Lockard and Pollack that is useful for showing that certain functions have a continuous distribution function where classical methods sometimes fail. |
| title | A statistical investigation of a divisor-sum function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.05284 |