We set, for all real $t \geqslant 2$,
$$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$
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.