We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^2}$ is integrable on $[2, +\infty[$.
We set, for all real $t \geqslant 2$,
$$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$
Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^2}$ is integrable on $[2, +\infty[$.