# Zeta function (operator)

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 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**^{[1]} in terms of the eigenvalues of the operator 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.^{[2]}

## ReferencesEdit

- Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006),
*Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings*, Springer Monographs in Mathematics, New York, NY: Springer-Verlag, ISBN 0-387-33285-5, Zbl 1119.28005 - Fursaev, Dmitri; Vassilevich, Dmitri (2011),
*Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory*, Theoretical and Mathematical Physics, Springer-Verlag, p. 98, ISBN 94-007-0204-3

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