  We consider the 3D Schr\"odinger operator $H = H_0 + V$ where $H_0 =
(-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic
field of strength $b>0$, and $V$ is a short-range electric potential which
decays superexponentially with respect to the variable along the magnetic
field. We show that the resolvent of $H$ admits a meromorphic extension from
the upper half-plane to an appropriate complex manifold ${\mathcal M}$, and
define the resonances of $H$ as the poles of this meromorphic extension. We
study their distribution near any fixed Landau level $2bq$, $q \in {\mathbb
N}$. First, we obtain a sharp upper bound of the number of resonances in a
vicinity of $2bq$. Moreover, under appropriate hypotheses, we establish
corresponding lower bounds which imply the existence of an infinite number of
resonances, or the absence of resonances in certain sectors adjoining $2bq$.
Finally, we deduce a representation of the derivative of the spectral shift
function (SSF) for the operator pair $(H,H_0)$ as a sum of a harmonic measure
related to the resonances, and the imaginary part of a holomorphic function.
This representation justifies the Breit-Wigner approximation, implies a trace
formula, and provides information on the singularities of the SSF at the Landau
levels.
