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How to find these limits
$\displaystyle\lim_{n\to\infty}\left(\ln(\ln(n)) - \sum_{k=2}^n\frac1{k \ln(k)}\right)$ ?

and $\displaystyle\lim_{n\to\infty}\left( \ln(\ln(n)) - \sum_{k=1}^n\frac1{p_k}\right)$?

where $p_k$ is the $k$'th prime number.

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    slightly related to http://math.stackexchange.com/questions/79115/limit-lim-limits-n-rightarrow-infty-left2-sqrt-n-sum-limits-k-1n-frac2011-11-05

2 Answers 2

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The second limit is precisely Mertens Constant.

The constant of the first limit, lets call it $C_{-1}$. I am not sure if it has a name. I believe Ramanujan computed that it was approximately $\approx 0.7946786$. See page 11 of this PDF for more details.

Remarkably it also appears in the following limit due to Ramanujan:

$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=2}^\infty \frac{1}{k(k^{\frac{1}{n}}-1)}-\log n=C_{-1}.$

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    I think what I missed is that Boas is talking about $\int_2^n(dx/x\log x)$ while we're talking about $\log\log n$, and they differ by $\log\log2$.2011-11-07
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Mertens proved the existence of $\lim(\sum_{p\le n}(1/p)-\log\log n)$ see here for more detail, or any good textbook for a proof. This isn't quite what's wanted in the second problem above, but it should get you started.