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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the irreducibility of the Dirichlet polynomial of an alternating group


Author: Massimiliano Patassini
Journal: Trans. Amer. Math. Soc. 365 (2013), 4041-4062
MSC (2010): Primary 11M41; Secondary 11N05, 20D06, 20E28
Published electronically: April 2, 2013
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Abstract: Given a finite group $ G$ the Dirichlet polynomial of $ G$ is

$\displaystyle P_{G}(s)=\sum _{H\leq G} \frac {\mu _G(H)}{\vert G:H\vert^s},$

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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Additional Information

Massimiliano Patassini
Affiliation: Dipartimento di Matematica, Università di Padova, Via Trieste, 63 - 35121 Padova, Italia
Email: frapmass@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05655-3
PII: S 0002-9947(2013)05655-3
Received by editor(s): May 12, 2011
Received by editor(s) in revised form: June 11, 2011, and June 19, 2011
Published electronically: April 2, 2013
Article copyright: © Copyright 2013 American Mathematical Society