SOME PROPERTIES ON THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial $P_G(s)$ associated with a finite group $G$ factorizes and the structure of G. If $P_G(s)$ is irreducible, then $G/FratG$ is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime $p\geq 5$ we show that if $P_G(s) = P_{Alt(p)}(s)$, then $G/FratG \cong Alt(p)$ and $P_{Alt(p)}(s)$ is irreducible. Moreover, if $P_G(s) = P_{PSL(2,p)}(s)$, then $G/FratG$ is simple, but $P_{PSL(2,p)}(s)$ is reducible whenever $p = 2^t- 1$ and $t = 3 mod 4$.