For \(T \gg T_{F}\), we get

\[
\frac{C_{\mathrm{V}}}{N k} \simeq \frac{n}{s}, \frac{C_{P}-C_{\mathrm{V}}}{N k} \simeq 1, \quad \text { so that } \frac{C_{P}}{N k} \simeq\left(\frac{n}{s}+1 \right)
\]

For \(T \ll T_{F}\), we obtain [see formula (E.17)]

\[
\begin{aligned}
\frac{C_{\mathrm{V}}}{N k} & =\frac{n}{s} \ln z\left\{1+\frac{n}{s} \frac{\pi^{2}}{3}(\ln z)^{-2}+\ldots \right\}-\frac{n}{s} \ln z\left\{1+\left(\frac{n}{s}-1 \right) \frac{\pi^{2}}{3}(\ln z)^{-2}+\ldots \right\} \\
& =\frac{n}{s} \frac{\pi^{2}}{3}(\ln z)^{-1}+\ldots \simeq \frac{n}{s} \frac{\pi^{2}}{3}\left(\frac{k T}{\varepsilon_{F}} \right) .
\end{aligned}
\]

To this order of accuracy, the quantity \(C_{P} / N k\) has the same value as \(C_{\mathrm{v}} / N k\). As for the difference between the two, we obtain

\[
\frac{C_{P}-C_{\mathrm{V}}}{N k} \simeq \frac{n}{s} \frac{\pi^{4}}{9}\left(\frac{k T}{\varepsilon_{F}} \right)^{3}
\]

consistent with the corresponding value of \(\gamma\) quoted in the previous problem. The non-relativistic case pertains to \(s=2\) while the extreme relativistic one pertains to \(s=1\).