Asymptotic relation between eigenvalue sum and chemical potential for electrons moving in bare point-charge potentials in $d$ dimensions

For electrons moving independently in a $d$-dimensional bare point-charge potential, the Euler equation of density-functional theory in the limit of large numbers of electrons, $N$, is combined with the virial theorem to relate the sum of eigenvalues $E$ to the chemical potential $\mu$. The result is $\frac{E}{N\mu }=\frac{d(4-d)\mathrm{}}{(4+2d-d^2)}\equiv {\alpha }^{-1\mathrm{}}$. This result (I) is verified by direct calculation of both $E$ and $\mu$ for $d=1,2, \mathrm{and} 3$ in the asymptotic limit of large $N$. Neither does there seem to be any difficulty in applying (I) for $d\ge 5$. But evidently $\frac{E}{N\mu }=0\mathrm{}$ for $d=4\mathrm{}$; an immediate consequence of the virial theorem for this dimensionality. Viewed as a continuous function of $d$, $\frac{E}{N\mu }$ is singular at $d=1+\sqrt{5}$ and is negative for $1+\sqrt{5}<d<\mathrm{}4\mathrm{}$. Excluding this pathological regime of $d$, it can be shown from (I) that, for a point charge $\mathrm{Ze}$, $E=F(Z,d)\mathrm{}{N}^{\alpha }$, where the exponent $\alpha$ is given by (I).