Fermi-Dirac half-order integral at zero argument
= Fermi-Dirac half-order integral at zero argument
{c}
Expanding the Fermi denominator as an alternating exponential series gives $F_{1/2}(0)=(1-2^{-1/2})\zeta(3/2)\simeq0.765147$. Thus a band with <chemical potential> pinned exactly at its lower edge has $n=F_{1/2}(0)N_c$, rather than the classical result $N_c$.