Distributions of discriminants of cubic algebras II

Abstract

Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spaces. In this paper we consider these objects under imposing finite number of splitting conditions at non-archimedean places. Especially the explicit formulae of residues at $s=1$ and 5/6 under the conditions are given.