On sums of monotone functions over smooth numbers

In this article, we are going to look at the requirements regarding a monotone function f is an element of R -> R->= 0, and regarding the sets of natural numbers (A(i))(i=1)(infinity) subset of dmn(f), which requirements are sufficient for the asymptotic Sigma(n is an element of AN P(n)<= N theta f(n) similar to rho(1/theta) Sigma)n is an element of AN( f(n)) to hold, where N is a positive integer, theta is an element of (0, 1) is a constant, P(n) denotes the largest prime factor of n, and rho is the Dickman function.