With $p_n$ prime, does the constant, $C$, exist and have a name? $$\sum_{n=1}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_n} = C$$
If not,how about a constant and function for this?
$$\sum_{n=1}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{f(n)} = C$$
Assuming I understand where you're trying to put your parentheses (since you have $\frac1{p_n}$ on the LHS I presume it's trying to be 'captured' by the sum), then the sum should be easily relatable to the Mertens constant : $\sum(\log\log p_{n+1}-\log\log p_n)$ telescopes, so if you sum from $n=1$ to $n=K$ your partial sum will be just $\log\log p_{K+1}-\log\log p_1$; this is approximately $\log\log(K\log K)$ (minus the starting term), and it's easy to show that this is $\log\log K+o(1)$. Meanwhile, $\sum_{n=1}^{K}\frac{1}{p_n}$ will grow as $\log\log K$, and the Mertens constant is the $O(1)$-sized difference between the two. Essentially, replacing $\log\log K$ with $\log\log p_K$ only adds a $o(1)$-sized term, not a constant-sized one.