Introduction to the Dickman Distribution

: The Dickman function ρ is continuous on [0,∞), ρ(x) = 1 for x ∈ [0, 1] and decreasing to 0 on [1,∞). It was introduced  by Dickman (1930) in the context of y-smooth numbers (numbers whose prime-factors  are no greater that y) and their asymptotic frequency. With a proper constant c, the function cρ(·) is the density function of the Dickman distribution. It is an infinitly divisible distribution. We shall discuss its properties, its generalizations and related statistical models e.g. perpetuities (a financial concept), records-low and more.