  We study the ground state of $N \gg 1$ noninteracting fermions in a
two-dimensional harmonic trap rotating at angular frequency $\Omega>0$. The
support of the density of the Fermi gas is a disk of radius $R_e$. We calculate
the variance of the number of fermions ${\cal N}_R$ inside a disk of radius $R$
centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and
interesting behaviours in two different scaling regimes: (i) $\Omega / \omega
<1 $ and (ii) $1 - \Omega / \omega = O(1/N)$, where $\omega$ is the angular
frequency of the oscillator. In the first regime (i) we find that ${\rm
Var}\,{\cal N}_{R}\simeq\left(A\log N+B\right)\sqrt{N}$ and we calculate $A$
and $B$ as functions of $R/R_e$, $\Omega$ and $\omega$. We also predict the
higher cumulants of ${\cal N}_{R}$ and the bipartite entanglement entropy of
the disk with the rest of the system. In the second regime (ii), the mean
fermion density exhibits a staircase form, with discrete plateaus corresponding
to filling $k$ successive Landau levels, as found in previous studies. Here, we
show that ${\rm Var}\,{\cal N}_{R}$ is a discontinuous piecewise linear
function of $\sim (R/R_e) \sqrt{N}$ within each plateau, with coefficients that
we calculate exactly, and with steps whose precise shape we obtain for any $k$.
We argue that a similar piecewise linear behavior extends to all the cumulants
of ${\cal N}_{R}$ and to the entanglement entropy. We show that these results
match smoothly at large $k$ with the above results for $\Omega/\omega=O(1)$.
These findings are nicely confirmed by numerical simulations. Finally, we
uncover a universal behavior of ${\rm Var}\,{\cal N}_{R}$ near the fermionic
edge. We extend our results to a three-dimensional geometry, where an
additional confining potential is applied in the $z$ direction.
