For all real $t \geqslant 2$, we define
$$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$
Establish that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} \underset{n \rightarrow +\infty}{=} \ln_{2}(n) + c_{1} + O\left(\frac{1}{\ln(n)}\right)$, for a real $c_{1} \in \mathbb{R}$ to be determined.