Natural Boundaries III: Nilpotent Groups

In the previous chapter, we found that Hypotheses 1 and 2 always held and the natural boundary for the zeta function of each algebraic group was an integer. Hence Corollary 5.9 ensured the existence natural boundary with no need to assume the Riemann Hypothesis. In this chapter, we consider the zeta functions of nilpotent groups and Lie rings listed in Chap. 2. We are not so lucky this time, since the candidate natural boundary is frequently nonintegral, and in many cases the existence of the natural boundary requires us to assume the Riemann Hypothesis.We find that Hypotheses 1 and 2 continue to hold for all calculated examples, although there seems to be no good reason why these hypotheses should hold in general.