On the irreducibility of the dirichlet polynomial of a simple group of lie type

Given a finite group G and a normal subgroup N of G, the Dirichlet polynomial of G given G/N is $${P_{G,N}}(s) = \sum\limits_{\scriptstyle H \le G \hfill \atop \scriptstyle NH = G \hfill} {{{{\mu _G}(H)} \over {|G:H{|^s}}}} .$$ In this paper, we assume that G is a primitive monolithic group with nonabelian socle soc(G) ≅ S n for some simple group S of Lie type. Under some assumptions on the Lie rank of S, we prove that P G,soc(G)(s) is irreducible in the ring of finite Dirichlet series. Moreover, we show that the Dirichlet polynomial P S (s) = P S,S (s) of a simple group S of Lie type is reducible if and only if S is isomorphic to A 1(p), where p is a Mersenne prime such that log2(p + 1) ≡ 3 (mod 4).