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We calculate the shear (eta) and bulk (zeta) viscosities of a weakly coupledquark gluon plasma at the leading-log order with finite temperature T and quarkchemical potential mu. We find that the shear viscosity to entropy densityratio eta/s increases monotonically with mu and eventually scales as (mu/T)^2at large mu. In contrary, zeta/s is insensitive to mu. Both eta/s and zeta/sare monotonically decreasing functions of the quark flavor number N_f when N_f\geq 2. This property is also observed in pion gas systems. Our perturbativecalculation suggests that QCD becomes the most perfect (i.e. with the smallesteta/s) at mu=0 and N_f = 16 (the maximum N_f with asymptotic freedom). It wouldbe interesting to test whether the currently smallest eta/s computed close tothe phase transition with mu=0 and N_f = 0 can be further reduced by increasingN_f.

Shear and Bulk Viscosities of a Weakly Coupled Quark Gluon Plasma with Finite Chemical Potential and Temperature---Leading-Log Results