Zeta functions of Lie rings of upper-triangular matrices

We prove a two-part theorem on local ideal zeta functions ofLie rings of upper-triangular matrices. First, we prove thatthese local zeta functions display a strong uniformity. Secondly,we prove that these zeta functions satisfy a local functionalequation. Some explicit examples of these zeta functions arealso presented. Finally, we consider certain quotients of theseLie rings, showing that the strong uniformity continues to hold,and that under certain circumstances the functional equationdoes too.