= Solution
In three dimensions use the $k$-space <density of states> $(L/2\pi)^3$ per <spin> species and the <spherical shell> volume $4\pi k^2dk$:
$$
g_3^{(1)}(\epsilon)=\left(\frac{L}{2\pi}\right)^3 4\pi k^2\frac{dk}{d\epsilon}
=\boxed{\frac{L^3}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2}.}
$$
Thus the <three-dimensional free-electron density of states> grows as $\epsilon^{1/2}$. For physical <Electrons>, including both <spin> states gives
$$
g_3(\epsilon)=\frac{L^3}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2}.
$$
All three total <density of states> functions have units $\mathrm{J^{-1}}$ when <energy> is in <joules>; they count states per energy interval. The paper explicitly suppresses <spin> in (a) but does not repeat that convention in (b), so giving both conventions avoids a hidden factor-of-two ambiguity. Parts (c)--(f) below use the physical two-spin-state <electron gas>.
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