  The homogeneous electron gas is one of the most studied model systems in
condensed matter physics. It is also at the basis of the large majority of
approximations to the functionals of density functional theory. As such, its
exchange-correlation energy has been extensively studied, and is well-known for
systems of 1, 2, and 3 dimensions. Here, we extend this model and compute the
exchange and correlation energy, as a function of the Wigner-Seitz radius
$r_s$, for arbitrary dimension $D$. We find a very different behavior for
reduced dimensional spaces ($D=1$ and 2), our three dimensional space, and for
higher dimensions. In fact, for $D > 3$, the leading term of the correlation
energy does not depend on the logarithm of $r_s$ (as for $D=3$), but instead
scales polynomialy: $ -c_D /r_s^{\gamma_D}$, with the exponent
$\gamma_D=(D-3)/(D-1)$. In the large-$D$ limit, the value of $c_D$ is found to
depend linearly with the dimension. In this limit, we also find that the
concepts of exchange and correlation merge, sharing a common $1/r_s$
dependence.
