  If the dimension $d$ is even, the resonances of the Schr\"odinger operator
$-\Delta +V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are
points on $\Lambda$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We
show that for fixed sign potentials $V$ and for nonzero integers $m$, the
resonance counting function for the $m$th sheet of $\Lambda$ has maximal order
of growth.
