This question already has an answer here:

Are upper/lower bounds known for the following quantity?

$$S(n,a)\stackrel{\triangle}{=}\sum_{p_k \leq n}\frac{1}{(\log p_k)^a}.$$

I am mainly interested in the case, $a=1$. I suppose with the approximation $p_k\approx k \log k$ it is going to be roughly equal to the logarithmic integral, but I hope that there is something better known to the experts.

In fact, if that's the best known result, how would one go about proving rigorously that it is asymptotic to $\mathrm{li}(n)$?