J Lagarias
A Two Variable Zeta Function for Number Fields
Abstract:
Based on an analogy with Arakelov
geometry, G. van der Geer and R. Schoof attached
a two-variable zeta function Z_K(w, s) to an algebraic
number field K, which for w = 1 specializes to the
completed Dedekind zeta function of K. This function
analytically continues as a meromorphic function of
two complex variables. This talk describes the
interpretation of this function in terms of Arakelov divisors,
and describes properties of the zero set of
this function in the case K=Q. For certain fields
this function is associated to a semigroup of
infinitely divisible probability densities on the
real line. More recent developments will be mentioned
as time permits.
(This is joint work with Eric Rains (IDA) and a preprint
appears at arXiv:math.NT/0104176)
