  We study the Ginibre ensemble of $N \times N$ complex random matrices and
compute exactly, for any finite $N$, the full distribution as well as all the
cumulants of the number $N_r$ of eigenvalues within a disk of radius $r$
centered at the origin. In the limit of large $N$, when the average density of
eigenvalues becomes uniform over the unit disk, we show that for $0<r<1$ the
fluctuations of $N_r$ around its mean value $\langle N_r \rangle \approx N r^2$
display three different regimes: (i) a typical Gaussian regime where the
fluctuations are of order ${\cal O}(N^{1/4})$, (ii) an intermediate regime
where $N_r - \langle N_r \rangle = {\cal O}(\sqrt{N})$, and (iii) a large
deviation regime where $N_r - \langle N_r \rangle = {\cal O}({N})$. This
intermediate behaviour (ii) had been overlooked in previous studies and we show
here that it ensures a smooth matching between the typical and the large
deviation regimes. In addition, we demonstrate that this intermediate regime
controls all the (centred) cumulants of $N_r$, which are all of order ${\cal
O}(\sqrt{N})$, and we compute them explicitly. Our analytical results are
corroborated by precise "importance sampling" Monte Carlo simulations.
