Zeta function (operator)

The zeta function of a mathematical operator

The zeta function may also be expressible as a spectral zeta function[1] in terms of the eigenvalues

by It is used in giving a rigorous definition to the functional determinant of an operator, which is given by

The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold.

One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.