S-duality and Strong Coupling Behavior of Large N Gauge Theories with N=4 Supersymmetry

We analyze the strong coupling behavior of the large N gauge theories in 4-dimensions with N=4 supersymmetry by making use of S-duality. We show that at large values of the coupling constant $\lambda=g_{YM}^2N$ the j-th non-planar amplitude $f_j(\lambda) (j=0,1,2 ...)$ behaves as $f_j(\lambda)\approx \lambda^j$. Implication of this behavior is discussed in connection with the supergravity theory on $AdS_5\times S^5$ suggested by the CFT/AdS correspondence. S-duality of the gauge theory corresponds to the duality between the closed and open string loop expansions in the gravity/string theory.