Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank.
type: Item
instance of: scholarly article
title: Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank.
zbMATH Open document ID: 1117.20021
published in: Journal of Pure and Applied Algebra
publication date: 2007-01-09T00:00:00Z
review text: 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.
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 5083187
zbMATH Keywords: multiple zeta functions
zbMATH Keywords: free Abelian groups
zbMATH Keywords: Dedekind domains
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: random finite quotients
author: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.05.010
OpenAlex ID: W2049336828
cites work: Q4146776
cites work: Q5563439
cites work: On the number of subgroups of finite abelian groups
cites work: On the asymptotic behaviour of the number of subgroups of finite abelian groups
cites work: Zeta function of subgroups of abelian groups and average orders
cites work: Zeta functions of arithmetic orders and Solomon's conjectures
cites work: A Unimodality Result in the Enumeration of Subgroups of a Finite Abelian Group
cites work: Q5849928
cites work: Q4004303
cites work: Subgroups of finite index in nilpotent groups
cites work: Subgroups of Finite Index in Free Groups
cites work: Introduction to algebra. Transl. from the Russian by Neal Koblitz
cites work: Inertia characteristics of self-adjoint matrix polynomials
cites work: On zeta-functions of finite abelian groups
cites work: On groups of polynomial subgroup growth
cites work: Subgroup growth.
cites work: Q3947818
cites work: Analytic properties of zeta functions and subgroup growth
cites work: On the Growth of Ideals and Submodules
cites work: Q4344656
cites work: Q3748279
cites work: Binomial posets, Möbius inversion, and permutation enumeration
cites work: Q4324348
DOI: 10.1016/J.JPAA.2006.05.010
description: scientific article; zbMATH DE number 5083187