Zeta function (operator)

The zeta function of a mathematical operator $$\mathcal O$$ is a function defined as


 * $$ \zeta_{\mathcal O}(s) = \operatorname{tr} \; \mathcal O^{-s} $$

for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.

The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues $$\lambda_i$$ of the operator $$\mathcal O$$ by


 * $$ \zeta_{\mathcal O}(s) = \sum_{i} \lambda_i^{-s} $$.

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


 * $$ \det \mathcal O := e^{-\zeta'_{\mathcal O}(0)} \;. $$

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.