Zeta function (operator)

The zeta function of a mathematical operator ${𝒪}$ is a function defined as

${\zeta }_{{𝒪}}(s)=\mathrm{tr}{{𝒪}}^{-s}$

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 ${𝒪}$ by

${\zeta }_{{𝒪}}(s)=\sum _i{\lambda }_i^{-s}$.

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

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

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.

See also

• Quillen metric

References

• 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 
• 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 978-94-007-0204-2