Computation of Partial Zeta Values ats=0 over a Totally Real Cubic Field

Abstract

In theory the problem of computing the special values of partial zeta functions of a totally real field was solved by Shintani. In practice it rarely works as the fundamental domain of the action of the relevant unit group is too cumbersome to apply in a computational setting. The object of this article is to give a computationally efficient version of Shintani's approach in the cubic case. The basic idea is to take the convex closure of the lattice points, in the first octant, and use the boundary points, of this convex set, to construct a more amenable fundamental domain. Thus the basic thrust is to arrive at an algorithm to find points on this boundary. For us the case of interest is whens=0. In this case we find a cubic analogue of the CF-formula for real quadratic number fields.