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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?

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  • $\begingroup$ From the PNT by induction $\sum_{n \le x, \Omega(n)=k} 1 \sim \frac{x}{\ln x} \frac{(\ln \ln x)^{k-1}}{(k-1)!}$ $\endgroup$ – reuns Apr 3 at 21:40
  • $\begingroup$ @reuns, so $Q_k(N)$ is not constant, $Q_k(N) \sim \alpha (k) N $ as $N \rightarrow \infty$, where $ \alpha (k) \rightarrow 0$ as $k\rightarrow \infty$ is this correct? $\endgroup$ – Aleksey Druggist Apr 4 at 10:54

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