  We state and give complete proof of the results of Siegel about the zeros of
the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the
importance of the determination of the limit to the left of the zeros of
$\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term
$-\sqrt{T/2\pi}P(\sqrt{T/2\pi})$ that would explain the periodic behaviour
observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$.
We precise also the connection of the position on the zeros of
$\mathop{\mathcal R}(s)$ with the zeros of $\zeta(s)$ in the critical line.
