Divisor Functions: Train-like Structure and Density Properties

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 $\zeta(s+1)$. The theoretical findings are supported by computations. Ironically, perfect and multiperfect numbers do not exhibit any distinctive characteristics for divisor functions.