# Bounds on sum of reciprocal of logarithm of primes [duplicate]

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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 26 '15 at 4:20

• 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). – GH from MO Aug 26 '15 at 4:31
• @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. – kodlu Aug 26 '15 at 4:47
• @GHfromMO:Regarding answer for $a>0$ in general, fair enough, thank you. – kodlu Aug 26 '15 at 5:18