= Solution
Let $g_v(E)$ denote the spin-summed <density of states> per volume. An electron moment with magnitude the <Bohr magneton> $\mu_B$ has <Zeeman energies> $\mp\mu_B B$. At low <temperature>, the two spin populations are
$$
n_\pm=\frac12\int_0^{\mu\pm\mu_B B}g_v(E)\,dE,
\qquad
n_+-n_-=g_v(E_F)\mu_B B+O(B^3).
$$
The common <chemical potential> changes only at order $B^2$, so it does not affect this linear imbalance. The <magnetization> is $M=\mu_B(n_+-n_-)=\mu_B^2g_v(E_F)B$. Define SI volume <magnetic susceptibility> by $M=\chi_p H$ and take the weak applied field as $B\simeq\mu_0H$. Then <Pauli paramagnetism> gives
$$
\boxed{\chi_p=\mu_0\mu_B^2g_v(E_F)=\frac{3\mu_0n\mu_B^2}{2E_F}.}
$$
The last equality uses the three-dimensional parabolic <free electron model>, $g_v(E_F)=3n/(2E_F)$. This susceptibility is nearly independent of <temperature> for $T\ll T_F$, unlike the <Curie law> for localized independent moments. The displayed result excludes orbital <diamagnetism> and interaction enhancements. If susceptibility instead denotes $\partial M/\partial B$, omit $\mu_0$; confusing that convention with $\partial M/\partial H$ changes the units.
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