Zeta function (operator) - Misplaced Pages

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.

References

Lapidus & van Frankenhuijsen (2006) p.23
Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, ISBN 0-521-41669-8, MR 1208731

Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006), Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics, New York, NY: Springer-Verlag, ISBN 0-387-33285-5, Zbl 1119.28005
Fursaev, Dmitri; Vassilevich, Dmitri (2011), Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Theoretical and Mathematical Physics, Springer-Verlag, p. 98, ISBN 978-94-007-0204-2

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