Less fundamental applications of Zeta regularization:

Question

As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect.

Are there less fundamental applications of zeta function regularization? By "less fundamental" I mean it 'naturally' pops up in more of an artificially / purely mathematically ideal constructed scenario.

Thanks!

Answer 1

Zeta-function regularization of the determinant of the Laplacian, for example on a torus, might qualify as a "purely mathematical" application.
See, for example, On functional determinants of Laplacians in polygons and simplicial complexes or Zeta functions and regularized determinants on projective spaces.

Answer 2

Zeta function regularization computes the asymptotics of smoothed sums.

https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

Also, the regularized determinant of the Laplacian is related to the Ray-Singer analytic torsion, which is equal to the Reidemeister torsion:

https://en.wikipedia.org/wiki/Analytic_torsion