This paper is devoted to the proof of the following estimate: Let n be even, \(n\geq 4\) and let \(V\in {\mathfrak C}_ 0^{\infty}({\mathbb{R}}^ n)\). Let S(\(\lambda)\) be the scattering matrix associated with the Schrödinger operator \(H=-\Delta +V\). It is known that S(\(\lambda)\) is a meromorphic function of Log \(\lambda\) on the Riemann surface of the logarithm. Then the number \(N(T^{\epsilon})\) of poles of S(\(\lambda)\) in the region \(T^{-\epsilon}\leq | \lambda | \leq T^{\epsilon},| \arg \lambda | \leq Log T^{\epsilon}\) is estimated, for any \(\epsilon \in [0,2^{-1/2})\) by \(N(T^{\epsilon})\leq C_{\epsilon}(1+T)^{n+1}\). That result extends an earlier result of a similar nature proved by Melrose for the odd dimensional case. The proof consists in relating the poles of S(\(\lambda)\) to those of \((\mathbf{1} -K(\lambda))^{-1}\) where K(\(\lambda)\) is a modified Lippmann Schwinger operator, by standard scattering theory arguments, of relating the poles of the latter to the zeroes of a suitable Fredholm determinant h(Log \(\lambda)\), of showing that h(z) is an entire function of z and of estimating its growth rate at infinity by the use of asymptotic estimates of the characteristic values of K(\(\lambda)\) for large \(\lambda\). The continuation of K(\(\lambda)\) and the latter estimates make use of the Hankel function representation of the free resolvent.