  We consider the so-called Dickman subordinator, whose Levy measure has
density 1/x restricted to the interval (0,1). The marginal density of this
process, known as the Dickman function, appears in many areas of mathematics,
from number theory to combinatorics. In this paper, we study renewal processes
in the domain of attraction of the Dickman subordinator, for which we prove
local renewal theorems. We then present applications to marginally relevant
disordered systems, such as pinning and directed polymer models, and prove
sharp second moment estimates on their partition functions.
