Euler number of instanton moduli space and Seiberg-Witten invariants

We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of the Euler number and the signature of the four manifolds vanishes. The partition function is the sum of the Euler number of instanton moduli space when it is possible to apply the vanishing theorem. Also we obtain a relation of the Euler number labeled by the instanton number k with Seiberg-Witten invariants. All calculations in this article are done without assuming duality.