Divisor Functions: Train-like Structure and Density Properties

We investigate the density properties of generalized divisor functions ${\displaystyle f_s(n)=\frac{\sum _{d\mathrm{\mid }n}{d}^s}{n^s}}$ and extend the analysis from the already-proven density of $s=1$ to $s\ge 0$. 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 $\mathrm{\mid }{f}_1(n)-a\mathrm{\mid }<\frac{1}{n^{1-\epsilon }}$ has infinitely many solutions and prove it for points in the range of $f_s$. We establish that $f_s$ is dense for $0<s\le 1$ but loses density for $s>1$. As a result, in the latter case the graphs experience ruptures. We extend Wolke’s discovery ${\displaystyle \mathrm{\mid }{f}_1(n)-a\mathrm{\mid }<\frac{1}{n^{0.4-\epsilon }}}$ to all $0<s\le 1$. 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.