Random generation of the special linear group

It is well known that the proportion of pairs of elements of \operatorname {SL}(n,q) which generate the group tends to 1 as q^n\to \infty . This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step in our proof is an estimate for the average of (\operatorname {ord} g)^{-1} when g ranges over \operatorname {GL} (n,q), which may be of independent interest. We prove that this average is \displaystyle \exp (-(2-o(1)) \sqrt {n \log n \log q}).