Li$_{ν}$( z ), where z = e βµ is a fugacity, using ∫ d $^{d}$k = 2 π d/ $^{2}$/ Γ( d/ 2) ∫ dkk d − 1 and the integrals valid for Re( ν ) > 0. The result, as expected, is proportional to T/λ d $_{T}$: There are two important limits to consider. For Bose gases near Bose-Einstein condensation, physically the interesting limit is µ/T → 0. Since Li$_{ν}$(1) = ζ ( ν ), where ζ is Riemann’s zeta function, this leads us to define the scaling functions c$_{d}$ ( µ/T ): where we have used ζ (2) = π $^{2}$/ 6. With the above normalizations, c$_{d}$ = 1 for a single free boson when µ/T = 0. The above formulas are well-defined for fermions at zero chemical potential. Using one finds as µ/T → 0: It should be pointed out that the coefficients c$_{d}$ are analogous to the Virasoro central charge for relativistic systems in d = 1, as discussed in [27]. The other interesting limit is T/µ → 0, i.e. z → ∞ or z → 0, depending on the sign of the chemical potential. Here the scaling forms are naturally based on the 12 ∫ ∞ 0 dx zx ν − 1 e x − z = Γ( ν )Li$_{ν}$( z ) ∞ 0 dx zx ν 1 e x + z = − Γ( ν )Li$_{ν}$( − z ) (21) F = − sT ( mT 2 π ) d/ 2 Li$_{(}$$_{d}$$_{+2)}$$_{/}$$_{2}$( sz ) (22) F = − πmT 2 12 c$_{2}$ ( µ/T ) ( d = 2) (23) F = − ζ (5 / 2) m 3 / $^{2}$T 5 / 2 (2 π ) 3 / 2 c$_{3}$ ( µ/T ) ( d = 3) − Li$_{ν}$( − 1) = ( 1 − 1 2 ν − 1 ) ζ ( ν ) (24) c$_{2}$ = 1 2 , c$_{3}$ = 1 − 1 $_{2}$√$_{2}$ (free fermions) (25)