The zeta function of a mathematical operator is a function defined as


for those values of s where this expression exists, and as an analytic continuation

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

 thereof for those values of s where it does not. Here "tr" denotes a functional trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

.

The zeta function may also be expressible in terms of the eigenvalues  of the operator by
.

It is used in giving a rigorous definition to the functional determinant

Functional determinant

In mathematics, if S is a linear operator mapping a function space V to itself, it is sometimes possible to define an infinite-dimensional generalization of the determinant. The corresponding quantity det is called the functional determinant of S.There are several formulas for the functional...

 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.