We consider profinite groups, i.e. inverse limits of finite groups. Our results can also be applied to residually finite groups, by considering their profinite completions. A profinite group G is compact, so has a finite Haar measure, which we normalize, and consider G as a probability space. G is termed positively finitely generated, if for some k, the probability P(G,k) that k random elements generate G is positive. E.g. finitely generated pro-p groups are PFG, but non-abelian free profinite free groups are not. We will characterize PFG groups by their subgroup growth (i.e. the number of finite index subgroups), and give examples of such groups, e.g. prosolvable groups. In some cases, the function P(G,k) can be interpolated to an analytic function P(G,s), given by a Dirichlet series. The reciprocal of this function has some claim to being called the zeta function of G.

 
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