Divisor Functions: Train-like Structure and Density Properties

We investigate the density properties of generalized divisor functions fₛ(n)=∑d|ndˢnˢ and extend the analysis from the already-proven density of $s=1$ to s≥0. We demonstrate that for every $s>0$, fₛ is locally dense, revealing the structure of fₛ as the union of infinitely many $trains$ -- specially organized collections of decreasing sequences -- which we define. We analyze Wolke's conjecture that |f₁(n)-a|<1n1-ε has infinitely many solutions and prove it for points in the range of fₛ. We establish that fₛ is dense for 0<s≤1 but loses density for $s>1$. As a result, in the latter case the graphs experience ruptures. We extend Wolke's discovery |f₁(n)-a|<1n0.4-ε to all 0<s≤1. In the last section, we prove that the rational complement to the range of fₛ is dense for all $s>0$. Thus, the range of f₁ 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.