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

We define the multiple zeta function of the free Abelian group ZdZd as ζZd(s1,…,sd)=∑|Zd:H|<∞α1(H)−s1⋯αd(H)−sd,ζZd(s1,…,sd)=∑|Zd:H|<∞α1(H)−s1⋯αd(H)−sd, where Zd/H≅Cα1(H)⊕⋯⊕Cαd(H)Zd/H≅Cα1(H)⊕⋯⊕Cαd(H) is the canonical decomposition into cyclic factors, and αi+1(H)|αi(H)αi+1(H)|αi(H) for i=1,…,d−1i=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 Zd/HZd/H as follows. For a subgroup of finite index H⊆ZdH⊆Zd, consider the order of the product of the canonical cyclic factors except the largest one, σ(H)=α2(H)⋯αd(H)σ(H)=α2(H)⋯αd(H). Fix n∈Nn∈N, and let σn(d)σn(d) be the arithmetic mean of σ(H)σ(H) over all subgroups H⊆ZdH⊆Zd of index at most nn. We prove that there exists a limit limn→∞σn(d)limn→∞σn(d), and this number is bounded by 1.243, for all ranks d≥1d≥1. In this sense, a random finite factor group Zd/HZd/H is very close to a cyclic group.We also compute the zeta function that enumerates cyclic finite factor groups. This result allows us to amend our observation that a random finite factor group Zd/HZd/H is close to a cyclic group in the following way. Consider all subgroups H⊆ZdH⊆Zd of index at most nn, and let τn(d)τn(d) be the share of the subgroups such that Zd/HZd/H is cyclic. We compute τ(d)=limn→∞τn(d)τ(d)=limn→∞τn(d), which can be considered as the probability that Zd/HZd/H is cyclic, and we show that τ(d)≥0.8469…τ(d)≥0.8469… for all d≥1d≥1.Also, we apply our results to study similar questions for free modules of finite rank over finitely generated Dedekind domains.