Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank.
Let \(\mathbb{Z}^d\) be the free Abelian group of rank \(d\) and let \(H\) be a subgroup of finite index. One can consider the canonical decomposition \(\mathbb{Z}^d/H\cong C_{\alpha_1(H)}\oplus\cdots\oplus C_{\alpha_d(H)}\) into cyclic factors, where \(\alpha_{i+1}(H)\) divides \(\alpha_i(H)\) for \(1\leq i\leq d-1\). This allows to define the multiple zeta function of \(\mathbb{Z}^d\) as follows: \(\zeta_{\mathbb{Z}^d}(s_1,\dots,s_d)= \sum_{|\mathbb{Z}^d:H|<\infty}\alpha_1(H)^{-s_1}\cdots\alpha_d(H)^{-s_d}\). The author computes this function, finds the region of absolute convergence and studies its analytic continuation. This allows to obtain interesting results on the asymptotic structure of a random finite factor group \(\mathbb{Z}^d/H\). For example, let \(\sigma(H)=\alpha_2(H)\cdots\alpha_d(H)\) (the order of the product of the canonical cyclic factors except the largest one), and, for a fixed \(n\in\mathbb{N}\), let \(\sigma_n(d)\) be the arithmetic mean of \(\sigma(H)\) over all subgroups \(H\leq\mathbb{Z}^d\) with \(|\mathbb{Z}^d:H|\leq n\); the author proves that \(\lim_{n\to\infty}\sigma_n(d)\) exists and is bounded by 1.243 (hence a random finite factor group \(\mathbb{Z}^d/H\) is very close to a cyclic group). The author computes also the zeta function that enumerates cyclic finite factor groups. In particular, he considers the number \(\tau_n(d)\) expressing the share of the subgroups \(H\leq\mathbb{Z}^d\) such that \(\mathbb{Z}^d/H\) is cyclic and proves that \(\lim_{n\to\infty}\tau_n(d)\geq 0.8469\). Similar questions are discussed for free modules of finite rank over finitely generated Dedekind domains.
scientific article; zbMATH DE number 5083187