  We calculate the free energy of a hot gas of electrons and photons to three
loops using the hard-thermal-loop perturbation theory reorganization of
finite-temperature perturbation theory. We calculate the free energy through
three loops by expanding in a power series in m_D/T, m_f/T, and e^2, where m_D
and m_f are thermal masses and e is the coupling constant. We demonstrate that
the hard-thermal-loop perturbation reorganization improves the convergence of
the successive approximations to the QED free energy at large coupling, e ~ 2.
The reorganization is gauge invariant by construction, and due to cancellation
among various contributions, we obtain a completely analytic result for the
resummed thermodynamic potential at three loops. Finally, we compare our result
with similar calculations that use the Phi-derivable approach.
