Show that the partition function (Q_{N}(V, T)) of an extreme relativistic gas consisting of (N) monatomic molecules

Question:

Show that the partition function $Q_{N}(V, T)$ of an extreme relativistic gas consisting of $N$ monatomic molecules with energy-momentum relationship $\varepsilon=p c, c$ being the speed of light, is given by

\[
Q_{N}(V, T)=\frac{1}{N !}\left\{8 \pi V\left(\frac{k T}{h c}ight)^{3}ight\}^{N}
\]

Study the thermodynamics of this system, checking in particular that

\[
P V=\frac{1}{3} U, \quad U / N=3 k T, \quad \text { and } \quad \gamma=\frac{4}{3}
\]

Next, using the inversion formula (3.4.7), derive an expression for the density of states $g(E)$ of this system.