Another enticing geometrophysical connection of the Bernoulli numbers from "Geometry and physics" by M. Atiyah, R. Dijkgraaf, and N. Hitchens, Phil. Trans. of the R. Soc., 368, 2010:

More recent developments have revolved around topological string theory on Calabi–Yau threefolds, relating them to invariants of sheaves and bundles. In string theory one considers not only rational curves, but Riemann surfaces
of general topology. This leads to generating functions $F_g$, where $g$ is the number of holes or genus of the surface, that generalize the function considered in §3. Out
of these one can make a master generating function,

$$Z = \exp\sum_{g≥0}λ^{2g−2}F_g,$$

that can be seen to count contributions of all possible topologies, including disconnected surfaces. The parameter λ is called the string coupling constant. These string theory invariants are a rich area for mathematical connections. In particular, for toric Calabi–Yau spaces they can be computed exactly.

There is remarkable re-interpretation of these string partition functions linking them directly to the so-called Donaldson–Thomas invariants of sheaves on the underlying Calabi–Yau space. Roughly, one re-expresses the function $Z$ in terms of a Fourier series

$$Z =\sum_d N_d q^d,$$

$q = e^\lambda\; $, where the coefficients $N_d$  now are assumed to equal the Donaldson–Thomas invariant of certain sheaves.

As an illustration we can consider the simplest possible Calabi–Yau manifold: complex three-space $C^3$. In this case the quantity $F_g$ can be computed (it is a rather complicated intersection number on the moduli space of Riemann surfaces) as

$$F_g = \frac{B_{2g}B_{2g−2}}{
2g(2g − 2)(2g − 2)!},$$

where the $B_n$ are the Bernoulli numbers ... .

Plugging these numbers into the expression for $Z$ gives a remarkable result

$$Z = \prod_{n>0}(1 − q^n)^{−n}.$$

This expression has a direct combinatorial interpretation in terms of a weighted sum over the so-called plane partitions that can be related to ideal sheaves on $C^3$.

(Note that $B_n/n = \zeta(1-n)$  for connections to the Riemann zeta function. Through the reflection formula this can also be connected to $\zeta(2n)$.)