  We prove that the resonance counting functions for Schr\"odinger operators
$H_V = - \Delta + V$ on $L^2 (\R^d)$, for $d \geq 2$ {\it even}, with generic,
compactly-supported, real- or complex-valued potentials $V$, have the maximal
order of growth $d$ on each sheet $\Lambda_m$, $m \in \Z \backslash \{0 \}$, of
the logarithmic Riemann surface. We obtain this result by constructing, for
each $m \in \Z \backslash \{0 \}$, a plurisubharmonic function from a
scattering determinant whose zeros on the physical sheet $\Lambda_0$ determine
the poles on $\Lambda_m$. We prove that the order of growth of the counting
function is related to a suitable estimate on this function that we establish
for generic potentials. We also show that for a potential that is the
characteristic function of a ball, the resonance counting function is bounded
below by $C_m r^d$ on each sheet $\Lambda_m$, $m \in \Z \backslash \{0\}$.
