The total number of particles is found from the grand potential by
This increases monotonically with z (up to the maximum z = +1). The behaviour when approaching z = 1 is however crucially dependent on the value of α (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).
For α > 1, the number of particles only increases up to a finite maximum value, i.e.,  is finite at z = 1:
where ζ(α) is the Riemann zeta function (using Li(1) = ζ(α)). Thus, for a fixed number of particles , the largest possible value that β can have is a critical value β. This corresponds to a critical temperature T=1/kβ, below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature:
For example, for the three-dimensional Bose gas in a box ( and using the above noted value of ) we get:
For α ≤ 1, there is no upper limit on the number of particles ( diverges as z approaches 1), and thus for example for a gas in a one- or two-dimensional box ( and  respectively) there is no critical temperature.