On sums of monotone functions over smooth numbers

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ _≥0, and regarding the sets of natural numbers (Ai)i=1∞⊆dmn(f) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑n∈ANP(n)≤Nθf(n)∼ρ(1/θ)∑n∈ANf(n) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.