Take $a_n\in 0,1$ such that $$\sum_{n\in  x} a_n=\frac{x}{\ln x}(1+\frac14 \cos(\ln x))+O(1)$$ then do a partial summation to find the asymptotic of $$\sum_{n\le x} \frac{a_n}{n}$$ And you meant $\sum_{p\le x} \frac1p= \ln\ln x + C + O(1/\ln x)$.