Zeta function (operator)

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

${\displaystyle \zeta _{\mathcal {O}}(s)=\operatorname {tr} \;{\mathcal {O}}^{-s}}$

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 function¹ in terms of the eigenvalues ${\displaystyle \lambda _{i}}$ of the operator ${\displaystyle {\mathcal {O}}}$ by

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

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

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

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.²