I've already asked a question regarding the Sum of almost-prime zeta functions. Now I'm interested in the next question, denote: $$\zeta_{k}^{al}(s, N)= \sum_{n=1}^{N} \frac{a(n)}{n^s},$$ where $$a_k(n)=1, \Omega(n)\leq k$$ $$a_k(n)=0, \Omega(n)>k$$
I'm interested in the following sum dependence on $N$ for big (and fixed) integer $k$:
$$Q_k(N)=\sum_{n=1}^N (1-a_k(n))$$
I gues for $N \leq e^{kln2}$ $Q_k(N)=0$, at $N \sim e^{e^k}$ the sum have inflection point (if we consider $N$ as a continuous variable) and for $N >> e^{e^k}$ it is constant. Is my guess correct?