Numbers whose prime divisors lie in special intervals

We study the distribution of numbers whose prime divisors lie in special intervals. Various multiplicative functions are summed over these numbers. For these summatory functions we obtain asymptotic formulae whose principal term is a sum of an increasing number of summands. We show that this sum can be approximated, up to the first rejected term, by a finite number of its summands. We also discuss relations on the parameters of the problem under which the principal term of such asymptotic formulae becomes a finite sum.