FWIW, I think the reason it does not show up on the "Bose-Einstein statistics" page: The zeta function is not so relevant to describing how the bosons populate

FWIW, I think the reason it does not show up on the "Bose-Einstein statistics" page: The zeta function is not so relevant to describing how the bosons populate states at particular energies, but zeta is relevant once you start summing/integrating over those populations (eg calculating expectation values.)