UC Santa Cruz

We prove a formula relating Dedekind zeta functions associated to a number field $k$ to certain Shintani zeta functions, whose analytic properties and values at non-positive integers have been well studied by Takuro Shintani. This allows us to compute explicit formulas for Dedekind zeta functions, partial zeta functions and certain $L$-series and their derivatives evaluated at non-positive integers. We relate the explicitly given value of the derivative of partial zeta functions at $s=0$ to those predicted by abelian Stark's conjecture. Though this conjecture remains open, we are able to write down explicit formulas for the absolute values of the conjectured Stark units.

The main ingredient in these formulas is an explicit proof of Shintani's unit theorem for number fields of arbitrary signature. This says that the totally positive units of a number field $k$ has a fundamental domain given by a signed union of polyhedral cones in the Minkowski space of the field. Existence of such domains was known to Shintani. In the case $k$ is a totally real field, Colmez, Diaz y Diaz--Friedman and Charollois-Dasgupta-Greenberg were able to construct such domains and give their generators explicitly. We give an explicit construction of such domains for number fields of arbitrary signature with an exact formula for the domain. Moreover, our construction is cohomological, allowing for future cohomological applications of Shintani's method as in the work of Charollois--Dasgupta--Greenberg.

This construction allows us to write Dedekind zeta functions and partial zeta functions in terms of certain analytic zeta functions defined over polyhedral cones (Shintani zeta functions). Thus we are able to translate questions about special values of Dedekind zeta functions to those about special values of Shintani zeta, whose values at non-positive integers are given by closed finite expressions due to work of Shintani.