| We study the function U (x, y, z) which is defined to be the number of positive integers not exceeding x which are free of prime divisors from the interval (z, y]. U (x, y, z) is closely related to well-known functions in analytic number theory, namely Y (x, z) = U (x, x, z) and F (x, y) = U (x, y, 1).;Dickman's function r (v) and Buchstab's function w (u) arise in the study of Y (x,z) and F (x,y), respectively.;As do r (v) and w (u) h (u, v) also satisfies difference-differential equations which allow us to study the asymptotic behaviour of h (u, v) as u, respectively v, approach infinity.;In the remainder of the thesis, we make use of the Laplace transform together with the saddle-point method to study h (u, v) as well as U (x, y, z). This method leads to a remarkable sharpening of the results on h (u, v) derived from the difference-differential equations. It also yields a more precise estimate for U (x, y, z) for a large domain in the (x, y, z)-space.;First, we deduce the following asymptotic formula for U (x, y, z), where we write u = log x/logy and v = logx/log z. We have, uniformly for x≥y≥z≥3/2 , Ux,y,z=xh u,v+Ox/log y, where hu,v is the continuous function defined by hu,v :=rv+ 0urtv/u wu-t dt, 0 |
