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Zeta function (operator)

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Source updated May 10, 2026

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.

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Zeta function (operator)

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