[ssivark]

> interesting to see the Riemann zeta function turn up here. Does anyone know why it cropped up here, in the critical temperature equation?

I don’t know if this is satisfactory, but here’s a technical answer: A lot of field theory calculations (when including 1-loop quantum effects) involve summing over all the modes/states in the field. In thermal equilibrium we can think of the time axis as a circle, and the wave frequencies get quantized into integers (upto a multiple of the temperature) aka Matsubara frequencies. So now, the calculation involves summing over all natural numbers (raised to some power), which is basically the zeta function. A more careful explanation can even justify the 3/2 (from the fact that we have 3 spatial dimensions, and a boson has action scaling as momentum squared) but it’s been a while since I’ve thought about this.