It is well known that the proportion of pairs of elements of
SL
(
n
,
q
)
\operatorname {SL}(n,q)
which generate the group tends to
1
1
as
q
n
→
∞
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
(
ord
g
)
−
1
(\operatorname {ord} g)^{-1}
when
g
g
ranges over
GL
(
n
,
q
)
\operatorname {GL} (n,q)
, which may be of independent interest. We prove that this average is
\[
exp
(
−
(
2
−
o
(
1
)
)
n
log
n
log
q
)
.
\exp (-(2-o(1)) \sqrt {n \log n \log q}).
\]