Minakshisundaram–Pleijel zeta function

Summary

The Minakshisundaram–Pleijel zeta function of a closed Riemannian manifold is the Dirichlet series $\sum \lambda_j^{-s}$ over the nonzero eigenvalues of its Laplacian. It has a pointwise version, with each term weighted by eigenfunctions. It converges for $\operatorname{Re} s > n/2$ and continues to a meromorphic function on $\C$. Its poles and special values carry geometric information, and its derivative at $0$ defines the zeta-regularized determinant of the Laplacian. It generalizes the Riemann zeta function, which arises from the circle.