Question

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$.

For example if $n = p^i$ for some prime $p$ then $D_n(x) \sim \log x/ \log p$.

What are the other 'nice' distribution function of divisors that are satisfied by an infinite family of integers ( like $\log x/ \log p$ for $\{p^i, i \in \mathbb{N}\}$ ).

Here 'nice', means it is easy to do calculus (integrate or differentiate) with such functions.

Answer 1

The distribution of divisors on 'average' is known (so to say giving the behavior of a 'typical'integer).

Set $F_n(u) = d(n)^{-1} D_n(n^u)$ where $d(n)$ is the total number of divisors of $n$, then

$$x^{-1} \sum_{n\le x} F_n(u) = \frac{2}{\pi} \arcsin \sqrt{u} + O((\log x)^{-1/2}) $$ uniformly for $x\ge 2$ and $u \in [0,1]$. This is due to Deshouillers, Dress, Tennenbaum (1979) and can be found for example in Tennenbaum's Introduction to Analytic and Probabilistic Number Theory (see Thm 7 in II.6).

Roughly this means, there are many small divisors.

A lot of information on this type of question, and I assume also this result but could not check, is in Hall and Tennenbaum's book 'Divisors'.