The linked article has proofs about the series 1/pi, but seems to only say that the divergence of the sum ∑ni=11/pi is greater than loglog(n+1).
I'd bet that limn→∞∑ni=11/pi∑ni=11/i→0
it seems obvious -- but I'm not sure how to prove that yet. Euler could probably have done it in his sleep. I'd need to mull over their approaches for a while.
It looks like this works
UPDATE: Yep, it's pretty obvious. The difference between the prime sum and log(log(n)) is finite, and so the numerator is close to log(log and the denominator to log so the ratio tends to 0. Anyhow, this Mathjax tool seems to pass the initial tests.
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