A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.