It's used in the computation for the Casimir effect (see

http://en.wikipedia.org/wiki/Casimir_effect).

It's use is somewhat explained in the last part of this article

http://plus.maths.org/content/infinity-or-just-112

Eh, that works because the Casimir energy is not the actual physical effect. The energy, at face value, is infinite as the photon modes that contribute to the energy are infinite. What is physical is the density of state of these modes with respect to the separation of the objects, which is taken as the spatial gradient (derivative) of the Casimir energy with respect to this separation. So while the energy is a divergent sum, its gradient is convergent because the higher order modes are of such small wavelength that they are not greatly affected by a small deviation in the scatterers (and thus the density of states remains unaffected as we go up in frequency). So we can renormalize the Casimir energy using any factor that we may wish as long as this factor drops out in the gradient. A typical factor is the energy when the objects are at infinite separation, which is what Casimir did in his original paper. In addition, he just windowed the frequencies by saying that beyond a certain frequency we have a cutoff function but this is unnecessary. With this treatment, you end up with an Euler-Maclaurin series that you can evaluate to a convergent sum. The zeta-regularization is another way of regularizing the divergent sum. Either way, it doesn't affect the actual physical result because the regularization drops out. Fortunately, you don't need to know a damn thing about zeta-regularization to be able to do a lot with Casimir theory. It actually is one of those nice quantum effects that exists on the macroscopic level so you can end up (through rigorous proofs) with a quasi-classical method of calculating the Casimir force.

This is an important part of Quantum Field Theory. In early Quantum Electrodynamics, one of the greatest problems they had were these divergencies and infinities. Regularization was an important breakthrough in removing these divergencies without affecting the final physics.

So yeah, they're doing some "don't look behind the curtain" hand waving here, but the subtle point is that the series 1+2+3+... is not the same series that we are truly evaluating when we say -1/12.