Dickman function

In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound.It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound.It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. The Dickman–de Bruijn function ρ ( u ) {displaystyle ho (u)} is a continuous function that satisfies the delay differential equation with initial conditions ρ ( u ) = 1 {displaystyle ho (u)=1} for 0 ≤ u ≤ 1. Dickman proved that, when a {displaystyle a} is fixed, we have where Ψ ( x , y ) {displaystyle Psi (x,y)} is the number of y-smooth (or y-friable) integers below x. Ramaswami later gave a rigorous proof that for fixed a, Ψ ( x , x 1 / a ) {displaystyle Psi (x,x^{1/a})} was asymptotic to x ρ ( a ) {displaystyle x ho (a)} , with the error bound in big O notation. The main purpose of the Dickman–de Bruijn function is to estimate the frequency of smooth numbers at a given size. This can be used to optimize various number-theoretical algorithms such as P-1 factoring and can be useful of its own right. It can be shown using log ⁡ ρ {displaystyle log ho } that which is related to the estimate ρ ( u ) ≈ u − u {displaystyle ho (u)approx u^{-u}} below.