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

ζ O ( s ) = tr O − s {\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 function in terms of the eigenvalues λ i {\displaystyle \lambda _{i}} of the operator O {\displaystyle {\mathcal {O}}} by

ζ O ( s ) = ∑ i λ i − s {\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

det O := e − ζ O ′ ( 0 ) . {\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.