概率zeta函数无负系数的投射有限群
Profinite groups in which the probabilistic zeta function has no negative coefficients
摘要：
对于一个随机生成的profinite群$G$，一个形式狄利克雷级数$P_G(s)=\sum_{n \in \mathbb N} {a_n(G)}/{n^s}$是相关联的，其中$a_n(G)=\sum_{|G:H|=n}μ(H, G)$和$μ(H,G)$表示$G.$的开子群格的莫比乌斯函数它的形式逆$P_G^{-1}(s)$是$G$的概率zeta函数。当$G$是前可解的时，$(P_G(s))^{-1}$的所有系数都是非负的.本文讨论了一般情形，给出了一个非预可解的例子的存在性，并构造了一个具有相同性质的非预可有限生成群$G$.
英文摘要：
To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)=\sum_{n \in \mathbb N} {a_n(G)}/{n^s}$ is associated, where $a_n(G)=\sum_{|G:H|=n}μ(H, G)$ and $μ(H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse $P_G^{-1}(s)$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of $(P_G(s))^{-1}$ is nonnegative. In this paper we discuss the general case and we produce % existence of a non-prosoluble example and We construct a non-prosoluble finitely generated group $G$ with the same property.
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