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The zeta function of a mathematical operator O {\displaystyle {\mathcal {O}}} is a function defined as
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
It is used in giving a rigorous definition to the functional determinant of an operator, which is given by
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