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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!

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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.

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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

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