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Natural Boundaries III: Nilpotent Groups

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1925)

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.

Keywords

  • Zeta Function
  • Algebraic Group
  • Nilpotent Group
  • Type IIIa
  • Natural Boundary

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2008 Springer-Verlag Berlin Heidelberg

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(2008). Natural Boundaries III: Nilpotent Groups. In: Zeta Functions of Groups and Rings. Lecture Notes in Mathematics, vol 1925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74776-5_7

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