It was not hard to google that the simpler sum $\displaystyle \sum_{p < N} \frac{1}{p} = c + \log \log N $ so it is divergent.
The sum of reciprocals of primes squared also converges $\displaystyle \sum_p \frac{1}{p^2} = 0.4522\dots$
Does $\displaystyle \sum_{p} \frac{1}{p} \left\{ \frac{N}{p} \right\} $ stay bounded as $N$ gets large? What about $\displaystyle \sum_{p,q } \frac{1}{pq} \left\{ \frac{N}{pq} \right\} $ for large $N$?
Maybe it's necessary to say $p,q < N$. This is related to my earlier question on the Euler $\phi$-function