Exact asymptotics of positive solutions to Dickman equation

Abstract

The paper considers the Dickman equation \begin{document}$\dot x (t)=-\frac{1}{t}\,x(t-1),$\end{document} for \begin{document} $t \to \infty $ \end{document} . The number theory uses what is called a Dickman (or Dickman -de Bruijn) function, which is the solution to this equation defined by an initial function \begin{document} $x(t)=1$ \end{document} if \begin{document} $0≤ t≤ 1$ \end{document} . The Dickman equation has two classes of asymptotically different positive solutions. The paper investigates their asymptotic behaviors in detail. A structure formula describing the asymptotic behavior of all solutions to the Dickman equation is given, an improvement of the well-known asymptotic behavior of the Dickman function, important in number theory, is derived and the problem of whether a given initial function defines dominant or subdominant solution is dealt with.