The density of orders of quotients of triangle groups

In this paper it is shown that for the natural density (among the positive integers) of the orders of the finite quotients of every ordinary triangle group is zero, using a modification of a component of a 1976 theorem of Bertram on large cyclic subgroups of finite groups, and the Turan-Kubilius inequality from asymptotic number theory. This answers a challenging question raised by Tucker, based on some work for special cases by May and Zimmerman and himself.