  The ring-diagram partial summation (or RPA) for the ground-state energy of
the uniform electron gas (with the density parameter $r_s$) in its
weak-correlation limit $r_s\to 0 $ is revisited. It is studied, which treatment
of the self-energy $\Sigma(k,\omega)$ is in agreement with the Hugenholtz-van
Hove (Luttinger-Ward) theorem $\mu-\mu_0= \Sigma(k_{\rm F},\mu)$ and which is
not. The correlation part of the lhs h as the RPA asymptotics $a\ln r_s
+a'+O(r_s)$ [in atomic units]. The use of renormalized RPA diagrams for the rhs
yields the similar expression $a\ln r_s+a''+O(r_s)$ with the sum rule $a'= a''$
resulting from three sum rules for the components of $a'$ and $a''$. This
includes in the second order of exchange the sum rule $\mu_{2{\rm
x}}=\Sigma_{2{\rm x}}$ [P. Ziesche, Ann. Phys. (Leipzig), 2006].
