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