Quotient order density of ordinary triangle groups

In this paper, we show that the natural density (among the positive integers) of the set of orders of the finite quotients of any ordinary triangle group is zero. This is achieved using methods inspired by a 1976 theorem of Bertram on large cyclic subgroups of finite groups and an application of the Turan-Kubilius inequality from asymptotic number theory. The question of the natural density of the set of finite quotient orders of triangle groups was first considered by Larson, then May and Zimmerman and then Tucker, who brought it to the author's attention.