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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 $\mathrm{li}(n)$?


marked as duplicate by GH from MO nt.number-theory Aug 26 '15 at 4:20

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    $\begingroup$ Just use the prime number theorem and partial summation. $\endgroup$ – Lucia Aug 26 '15 at 1:47
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    $\begingroup$ You could also e.g. split the sum at n/(log n)^100, I think. $\endgroup$ – alpoge Aug 26 '15 at 3:01
  • $\begingroup$ Note that $p_k$ is asymptotic to $k\log k$, not $k/\log k$. Also, there is no $a$ in your question, so the comment regarding the $a=1$ case makes no sense. At any rate, please open a new question if necessary (I marked this one as duplicate). $\endgroup$ – GH from MO Aug 26 '15 at 4:31
  • $\begingroup$ @GHfromMO: I understand why you marked as duplicate. Apologies, there was a typo, now fixed. Also, I had not found upon search the other answer, which answers the case $a=1.$ Any info on $a\neq 1$ would be appreciated. $\endgroup$ – kodlu Aug 26 '15 at 4:47
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    $\begingroup$ @GHfromMO:Regarding answer for $a>0$ in general, fair enough, thank you. $\endgroup$ – kodlu Aug 26 '15 at 5:18

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