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Let Z(X) be the number of degree-d extensions of F_q(t) with boundeddiscriminant and some specified Galois group. The problem of computing Z(X) canbe related to a problem of counting F_q-rational points on certain Hurwitzspaces. Ellenberg and Venkatesh used this idea to develop a heuristic for theasymptotic behavior of Z'(X), the number of -geometrically connected-extensions, and showed that this agrees with the conjectures of Malle forfunction fields. We extend Ellenberg-Venkatesh's argument to handle the morecomplicated case of covers of P^1 which may not be geometrically connected, andshow thatthe resulting heuristic suggests a natural modification to Malle'sconjecture which avoids the counterexamples, due to Kl\"uners, to the originalconjecture.

Connected Components of Hurwitz Schemes and Malle's Conjecture