Show that, in two dimensions, the specific heat \(C_{V}(N, T)\) of an ideal Fermi gas is identical to the specific heat of an ideal Bose gas, for all \(N\) and \(T\).

[It will suffice to show that, for given \(N\) and \(T\), the thermal energies of the two systems differ at most by a constant. For this, first show that the fugacities, \(z_{F}\) and \(z_{B}\), of the two systems are mutually related:

\[\left(1+z_{F}\right)\left(1-z_{B}\right)=1 \text {, i.e., } \quad z_{B}=z_{F} /\left(1+z_{F}\right) \text {. }\]

Next, show that the functions \(f_{2}\left(z_{F}\right)\) and \(g_{2}\left(z_{B}\right)\) are also related:

\[\begin{aligned}f_{2}\left(z_{F}\right) & =\int_{0}^{z_{F}} \frac{\ln (1+z)}{z} d z \\& =g_{2}\left(\frac{z_{F}}{1+z_{F}}\right)+\frac{1}{2} \ln ^{2}\left(1+z_{F}\right)\end{aligned}\]

It is now straightforward to show that

\[E_{F}(N, T)=E_{B}(N, T)+\text { const. },\]

the constant being \(E_{F}(N, 0)\).]