  We consider the Landau Hamiltonian perturbed by a long-range electric
potential $V$. The spectrum of the perturbed operator consists of eigenvalue
clusters which accumulate to the Landau levels. First, we obtain an estimate of
the rate of the shrinking of these clusters to the Landau levels as the number
of the cluster $q$ tends to infinity. Further, we assume that there exists an
appropriate $\V$, homogeneous of order $-\rho$ with $\rho \in (0,1)$, such that
$V(x) = \V(x) + O(|x|^{-\rho - \epsilon})$, $\epsilon > 0$, as $|x| \to
\infty$, and investigate the asymptotic distribution of the eigenvalues within
a given cluster, as $q \to \infty$. We obtain an explicit description of the
asymptotic density of the eigenvalues in terms of the mean-value transform of
$\V$.
