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

$det𝒪:=e^{-\zeta {\text{'}}_{𝒪}(0)}.$

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

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