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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.03497 |
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| _version_ | 1866917686215180288 |
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| author | Dubovski, Evelina |
| author_facet | Dubovski, Evelina |
| contents | We investigate the density properties of generalized divisor functions $\displaystyle f_s(n)=\frac{\sum_{d|n}d^s}{n^s}$ and extend the analysis from the already-proven density of $s=1$ to $s\geq0$. We demonstrate that for every $s>0$, $f_s$ is locally dense, revealing the structure of $f_s$ as the union of infinitely many $trains$ -- specially organized collections of decreasing sequences -- which we define. We analyze Wolke's conjecture that $|f_1(n)-a|<\frac{1}{n^{1-\varepsilon}}$ has infinitely many solutions and prove it for points in the range of $f_s$. We establish that $f_s$ is dense for $0<s\leq1$ but loses density for $s>1$. As a result, in the latter case the graphs experience ruptures. We extend Wolke's discovery $\displaystyle |f_1(n)-a|<\frac{1}{n^{0.4-\varepsilon}}$ to all $0<s\leq1$. In the last section, we prove that the rational complement to the range of $f_s$ is dense for all $s>0$. Thus, the range of $f_1$ and its complement form a partition of rational numbers to two dense subsets. If we treat the divisor function as a uniformly distributed random variable, then its expectation turns out to be $ζ(s+1)$. The theoretical findings are supported by computations. Ironically, perfect and multiperfect numbers do not exhibit any distinctive characteristics for divisor functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisor Functions: Train-like Structure and Density Properties Dubovski, Evelina Number Theory 11N64 We investigate the density properties of generalized divisor functions $\displaystyle f_s(n)=\frac{\sum_{d|n}d^s}{n^s}$ and extend the analysis from the already-proven density of $s=1$ to $s\geq0$. We demonstrate that for every $s>0$, $f_s$ is locally dense, revealing the structure of $f_s$ as the union of infinitely many $trains$ -- specially organized collections of decreasing sequences -- which we define. We analyze Wolke's conjecture that $|f_1(n)-a|<\frac{1}{n^{1-\varepsilon}}$ has infinitely many solutions and prove it for points in the range of $f_s$. We establish that $f_s$ is dense for $0<s\leq1$ but loses density for $s>1$. As a result, in the latter case the graphs experience ruptures. We extend Wolke's discovery $\displaystyle |f_1(n)-a|<\frac{1}{n^{0.4-\varepsilon}}$ to all $0<s\leq1$. In the last section, we prove that the rational complement to the range of $f_s$ is dense for all $s>0$. Thus, the range of $f_1$ and its complement form a partition of rational numbers to two dense subsets. If we treat the divisor function as a uniformly distributed random variable, then its expectation turns out to be $ζ(s+1)$. The theoretical findings are supported by computations. Ironically, perfect and multiperfect numbers do not exhibit any distinctive characteristics for divisor functions. |
| title | Divisor Functions: Train-like Structure and Density Properties |
| topic | Number Theory 11N64 |
| url | https://arxiv.org/abs/2406.03497 |