Great application of zeta.

This would be true for massive particles (not massless particles) in $3D$ space. Why $n=3/2$ is because it turns out that

$\frac { 2 }{ \sqrt { \pi } } \int _{ 0 }^{ \infty }{ \frac { \sqrt { x } }{ { e }^{ x }-1 } } dx=\zeta \left( \frac { 3 }{ 2 } \right)$

where the denominator relates to the distribution of possible states of bosons that can share the same space, and the numerator relates to the density of same in $3D$ space. This much-simplified explanation of Bose-Einstein statistics is to at least give an idea of why the Zeta function pops up.

Bosons are not always "massless particles", such as photons. Massive combinations of particles can have $0$ spin and thus behave like bosons.