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[1] in terms of the
eigenvalues of the operator by
.
It is used in giving a rigorous definition to the
functional determinant of an operator, which is given by
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]
^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,
ISBN0-521-41669-8,
MR1208731
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,
ISBN0-387-33285-5,
Zbl1119.28005
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,
ISBN978-94-007-0204-2