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The analytic equation of state of nonideal Coulomb plasmas consisting of pointlike ions immersed in a polarizable electron background [G. Chabrier and A. Y. Potekhin, Phys. Rev. E 58, 4941 (1998)] is improved, and its applicability range is considerably extended. First, the fit of the electron screening contribution in the free energy of the Coulomb liquid is refined at high densities where the electrons are relativistic. Second, we calculate the screening contribution for the Coulomb solid (bcc and fcc) and derive an analytic fitting expression. Third, we propose a simple approximation to the internal and free energy of the liquid one-component plasma of ions, accurate within the numerical errors of the most recent Monte Carlo simulations. We obtain an updated value of the coupling parameter at the solid-liquid phase transition for the one-component plasma: Gamma(m)=175.0+/-0.4(1sigma).

Equation of state of fully ionized electron-ion plasmas. II. Extension to relativistic densities and to the solid phase