Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

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.