It appears in some quantum field theory calculations, in particular for Casimir Effect.

In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate).

https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C...

and

https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...