Show that $\int_{\lfloor x \rfloor}^{x} \frac{q(u)}{u} \mathrm{~d}u$ tends to 0 when $x$ tends to $+\infty$, and deduce the convergence of the integral $\int_{1}^{+\infty} \frac{q(u)}{u} \mathrm{~d}u$, as well as the equality
$$\int_{1}^{+\infty} \frac{q(u)}{u} \mathrm{~d}u = \frac{\ln(2\pi)}{2} - 1$$