On the irreducibility of the Dirichlet polynomial of an alternating group

Patassini, Massimiliano

doi:10.1090/S0002-9947-2013-05655-3

On the irreducibility of the Dirichlet polynomial of an alternating group
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Trans. Amer. Math. Soc. 365 (2013), 4041-4062 Request permission

Abstract:

Given a finite group $G$ the Dirichlet polynomial of $G$ is \[ P_{G}(s)=\sum _{H\leq G} \frac {\mu _G(H)}{|G:H|^s},\] where $\mu _G$ is the Möbius function of the subgroup lattice of $G$. This object is a member of the factorial domain of finite Dirichlet series. In this paper we prove that if $G$ is an alternating group of degree $k$ and $k\leq 4.2\cdot 10^{16}$ or $k\geq (e^{e^{15}}+2)^3$, then $P_G(s)$ is irreducible. Moreover, assuming the Riemman Hypothesis, we prove that $P_G(s)$ is irreducible in the remaining cases.

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