Zeta Functions of Burnside Rings of Groups of Order p and p 2

The purpose of this article is to determine ζ B(G)(s) the zeta function of B(G) the Burnside ring, for the cyclic groups G = C p and C p 2 , where p is a prime integer, in order to see its behavior and make conjectures for further research. We study thoroughly all ideals of finite index in the ℤ(p) − order B (p)(G), to find ζ B (p)(G)(s). Furthermore, we compute the polynomial f (G)(p −s ), that belongs to ℤ[p −s ] and relates ζ B (p)(G)(s) to the zeta function of the maximal order