On the Distribution of Small Powers of a Primitive Root

Let Ng={gn:1⩽n⩽N}, where g is a primitive root modulo an odd prime p, and let fg(m, H) denote the number of elements of Ng that lie in the interval (m, m+H], where 1⩽m⩽p. H. Montgomery calculated the asymptotic size of the second moment of fg(m, H) about its mean for a certain range of the parameters N and H and asked to what extent this range could be increased if one were to average over all the primitive roots (modp). We address this question as well as the related one of averaging over the prime p.