The '''zeta function of a mathematical operator''' <math>\mathcal O</math> is a function defined as

:<math> \zeta_{\mathcal O}(s) = \operatorname{tr} \; \mathcal O^{-s} </math>

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'''<ref>Lapidus & van Frankenhuijsen (2006) p.23</ref> in terms of the eigenvalues <math>\lambda_i</math> of the operator <math>\mathcal O</math> by

:<math> \zeta_{\mathcal O}(s) = \sum_{i} \lambda_i^{-s} </math>.

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

:<math> \det \mathcal O := e^{-\zeta'_{\mathcal O}(0)} \;. </math>

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.<ref>{{citation | last=Soulé | first= C. |author-link=Christophe Soulé | title=Lectures on Arakelov geometry | series= Cambridge Studies in Advanced Mathematics | volume= 33 | publisher=Cambridge University Press | place=Cambridge | year= 1992 | pages= viii+177 | isbn= 0-521-41669-8 |mr=1208731 | others= with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer}}</ref>

== See also ==

* Quillen metric

==References== {{reflist}} * {{citation | last1=Lapidus | first1=Michel L. | last2=van Frankenhuijsen | first2=Machiel | title=Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings | series=Springer Monographs in Mathematics | location=New York, NY | publisher=Springer-Verlag | year=2006 | isbn=0-387-33285-5 | zbl=1119.28005 }} *{{citation|title=Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory |series=Theoretical and Mathematical Physics |first1=Dmitri |last1=Fursaev |first2=Dmitri |last2=Vassilevich |publisher=Springer-Verlag |year=2011 | isbn=978-94-007-0204-2 | page=98 }}

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Category:Functional analysis Category:Zeta and L-functions