From formal math perspective, Riemann Zeta function describes the probability density of prime number distribution in complex plane, i.e. when expressed by Ulam's spiral.
From AWT perspective Riemann Zeta function is significant from two dual reasons. It first, Ulam's spiral is closely related to Fibonacci spiral, which corresponds the patterns of closest packing of colliding particles and Golden mean ratio. Plant leaves and septum of snail/sea shells are minimizing the volume/surface ratio by the same way, like growing sphere formed by mutually coliding particles in diffusive (random walk) process.
Because numbers in vector space can be considered as a formal representation of colliding countable particles in N-dimensional space, we can see, how closely the number theory is related to particle physics and vice-versa.
Zeta function is therefore important in description of supergravity (SUGRA), which is general interaction between density gradients, formed by their shielding effect in N-dimensional space in analogy to Fatio de Duillier - Le Sage theory of gravity. The common gravity and Casimir force are projections of supergravity into 3D space from this perspective. The contributions of supergravity in higher dimensions converge to zero in accordance to surface/volume ratio dependence of N-dimensional hypersphere (i.e. hypersurface area) and the high dimensional residuals are promoting the complexity of observable reality.
The consequence is, inside of dense particle field the nested density fluctuations (clusters) are formed, but the number of cluster generations is strictly limited by geometry of their mutual interactions. After some three generations a less ore more chaotic particle superfluid is restored, as can be demonstrated by computer simulations. This behavior is related to many natural phenomena, from hydraulic sterilization over shaped warhead theory and viscoelastic properties of complex polymers (tixotropy of ketchup or Thinking Putty plasticine) and certain alloys (bismuth) under pressure to boson condensate and HT superconductivity theory. AWT explains the existence of three particle generations and the presence of superfluid condensate forming the vacuum inside of black holes by this way.