Is there a general formula for partial sum of reciprocals of square roots of k-almost primes?
This gives density of k-almost primes as $$\pi_k(x) \approx \frac{x}{\log{x}} \cdot \frac{({\log{\log{x}})^{k-1}}}{(k-1)!} $$
After a following the method - as suggested - in a related question about primes, I am now stuck at evaluating the integral expression in RHS.
Denoting $P_k$ as set of k-almost primes, $p_k$ as a member of set $P_k$, and $\rho_{\kappa}$ is the smallest member of $P_k$, we get:
$$\sum_{\rho_{\kappa}}^x \frac{1}{\sqrt{p_k}} = \int_{\rho_{\kappa}}^x \frac{d(\pi_k(t))}{\sqrt{t}} $$ $$ =\frac{\pi_k(t)}{\sqrt{t}} \Big \vert_{\rho_k}^x - \int_{\rho_{\kappa}}^x \pi_k(t) \frac{1}{-2\sqrt{t}} dt $$ $$= \frac{\sqrt{x}}{\log{x}} \cdot \frac{({\log{\log{x}})^{k-1}}}{(k-1)!} - \frac{\sqrt{\rho_k}}{ \log{\rho_k}} \cdot \frac{({\log{\log{\rho_k}})^{k-1}}}{(k-1)!} \\ + \frac{1}{2} \int_{\rho_{\kappa}}^x \frac{1}{\sqrt{t} \log{t}} \cdot \frac{({\log{\log{t}})^{k-1}}}{(k-1)!}dt $$
I am unsure about whether the following integral is convergent - and if it is - it would be nice to find out what it converges to:
$$ \int_{\rho_{\kappa}}^x \frac{({\log{\log{t}})^{k-1}}}{\sqrt{t} \log{t}} \cdot dt $$