Correlation energy of the spin-polarized uniform electron gas at high density

The correlation energy per electron in the high-density uniform electron gas can be written as $\Ec(r_s,\zeta) = \lam_0(\zeta) \ln r_s + \eps_0(\zeta) + \lam_1(\zeta) \,r_s \ln r_s + O(r_s)$, where $r_s$ is the Seitz radius and $\zeta$ is the relative spin polarization. We derive an expression for $\lam_1(\zeta)$ which is exact for any $\zeta$, including the paramagnetic and ferromagnetic limits, $\lam_1(0)$ and $\lam_1(1)$, and discover that the previously published $\lam_1(1)$ value is incorrect. We trace this error to an integration and limit that do not commute. The spin-resolution of $\lam_1(\zeta)$ into contributions of electron pairs is also derived.