On the irreducibility of the dirichlet polynomial of an alternating group

摘要

Given a finite group G the Dirichlet polynomial of G is P_G(s) = ∑~(H≤G)μ_G(H)/|G: H|~s, where μG is the M?obius 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 ≤ 4.2 · 10~(16) or k ≥ (e~(e15) + 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.