Zeta function (operator) explained

The zeta function of a mathematical operator

l O is a function defined as\zeta _{lO} (s) = \operatorname{tr} l 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.

λ _{i} of the operator l O by\zeta _{lO} (s) = \sum _{i} λ ^{- s} _{i} .

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

\det l O : = e ^{- \zeta' _{lO} ( 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.

See also

• Quillen metric

Notes and References

1. Lapidus & van Frankenhuijsen (2006) p.23