Around the infinite divisibility of the Dickman distribution and related topics

There are two probability distributions related to the Dickman function from number theory, which are sometimes confused with each other. We give a careful exposition on the difference between the two. While one is known to be infinite divisible, we give a computational proof to show that the other is not. We apply this to get related results for self-decomposable distributions with so-called truncated Lévy measures. Further, we extend several results about the infinitely divisible Dickman distribution related to its role in the context of sums on independent random variables and perpetuities. Along the way, we discuss several approaches for checking if a distribution is or is not infinitely divisible.