The Convolution Powers of the Dickman Function

Abstract

There is a family of functions wα(u), 0 < α < ∞, related to the Dickman function ρ(u) by the relations w1 = e−γ ρ(u), wα ∗ wrβ = wα+rβ(u). We consider the analytic properties of the wα(u) and obtain an asymptotic formula for wα(u) valid uniformly in α ⩾ ε, u+α → ∞. The wα(u) are of direct interest in number theory as well as for their connection to the Dickman function.