Abstract :
We define the multiple zeta function of the free Abelian group image as image where image is the canonical decomposition into cyclic factors, and αi+1(H)αi(H) for i=1,…,d−1. As the main result, we compute this function, find the region of absolute convergence, and study its analytic continuation.
Our result allows us to describe an asymptotic structure of a “random” finite factor group image as follows. For a subgroup of finite index image, consider the order of the product of the canonical cyclic factors except the largest one, σ(H)=α2(H)cdots, three dots, centeredαd(H). Fix image, and let σn(d) be the arithmetic mean of σ(H) over all subgroups image of index at most n. We prove that there exists a limit limn→∞σn(d), and this number is bounded by 1.243, for all ranks d≥1. In this sense, a random finite factor group image is very close to a cyclic group.
