Zeta function regularization

Summary

Zeta function regularization assigns finite values to divergent sums $\sum \lambda_n$ and divergent products $\prod \lambda_n$ of positive numbers. One forms the zeta function $\zeta_\lambda(s) = \sum \lambda_n^{-s}$, which converges for large $\operatorname{Re} s$, and continues it by analytic continuation. The regularized sum is $\zeta_\lambda(-1)$, and the regularized product is $e^{-\zeta_\lambda'(0)}$. For $\lambda_n = n$ the Riemann zeta function gives the regularized value $-\frac{1}{12}$ of $1 + 2 + 3 + \cdots$. The method is used for determinants of Laplacians and in quantum field theory.