De Bruijn 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, which is not easily available, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. Definition The Dickman–de Bruijn function \rho(u) is a continuous function that satisfies the delay differential equation :u\rho'(u) + \rho(u-1) = 0\, with initial conditions \rho(u) = 1 for 0 ≤ ''u'' ≤ 1. Properties Dickman proved that, when a is fixed, we have :\Psi(x, x^)\sim x\rho(a)\, where \Psi(x,y) is the number of ''y''-smooth (or ''y''-friable In materials science, friability ( ), the condition of being friable, describes the tendency of a solid substance to break into smaller pieces under stress or contact, especially by rubbing. The opposite of friable i ...
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