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zero temperature degenerate free fermion gas, where µ > 0 is the Fermi energy. In fact, the function Li$_{ν}$( z ) has a branch cut along the real axis from z = 1 to ∞ and the bosonic free energy is ill-defined at z = ∞ , in contrast with fermions. Using the analytic continuation of the asymptotic behavior 

<img src='content_image/338011.jpg'>

from positive integer ν to half-integer values, we define the scaling functions b$_{d}$ as follows: 

<img src='content_image/338012.jpg'>

The above normalizations are defined such that b$_{d}$ (0) = 1 for a single free fermion. One can verify that the above d = 3 expression is the standard result for a zero temperature, single component, degenerate fermion gas where µ is the Fermi energy. 

Other thermodynamic quantities follow from the free energy. The pressure p = −F . The density is n = − ∂ F /∂µ and the entropy density is s = − ∂ F /∂T . Using the scaling form in eq. (27), one obtains for d = 3: 

<img src='content_image/338013.jpg'>

where b ′ is the derivative with respect to its argument T/µ . (Henceforth, g ′ will always refer to the derivative of the function g with respect to it’s argument µ/T or T/µ as defined above.) The analogous formulas in 2 dimensions are: 

<img src='content_image/338014.jpg'>

