@BySymmetry okay thanks, I've just edited the original question to add more info on the problem I'm actually trying to solve.

Ok but I'm using $E=0$ for a single particle in the bottom of the potential well, so I can't use $E=0$ for having no particles at all. The choice of $E$ for no particles does matter when I try to compute $\langle n^b \rangle$ (the expected number of particles in system $b$).

What would the energy of this empty state be? i.e. $\sum_i g(E_i) exp[-\beta\,E_i] = g(E_0) exp[-\beta\,E_0] = exp[-\beta\,E_0] $.