Remarks on Two-Loop Free Energy in N=4 Supersymmetric Yang-Mills Theory at Finite Temperature

The strong coupling behavior of finite temperature free energy in N=4 supersymmetric SU(N) Yang-Mills theory has been recently discussed by Gubser, Klebanov and Tseytlin in the context of AdS-SYM correspondence. In this note, we focus on the weak coupling behavior. As a result of a two-loop computation we obtain, in the large N 't Hooft limit, $F(g^2N\to 0)\approx -\frac{\pi^2}{6}N^2V_3T^4(1-\frac{3}{2\pi^2}g^2N)$. Comparison with the strong coupling expansion provides further indication that free energy is a smooth monotonic function of the coupling constant.