zeta function (operator)

The zeta function of a mathematical operator $\backslash mathcal\; O$ is a function defined as

:$\backslash zeta\_\{\backslash mathcal\; O\}(s)\; =\; \backslash operatorname\{tr\}\; \backslash ;\; \backslash mathcal\; O^\{-s\}$

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 functionLapidus & van Frankenhuijsen (2006) p.23 in terms of the eigenvalues $\backslash lambda\_i$ of the operator $\backslash mathcal\; O$ by

:$\backslash zeta\_\{\backslash mathcal\; O\}(s)\; =\; \backslash sum\_\{i\}\; \backslash lambda\_i^\{-s\}$.

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

:$\backslash det\; \backslash mathcal\; O\; :=\; e^\{-\backslash zeta\text{'}\_\{\backslash mathcal\; O\}(0)\}\; \backslash ;.$

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.{{citation | last=Soulé | first= C. | 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 | author2= with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer}}