On the distribution of small powers of a primitive root

Let V-g={g(n) : 1 less than or equal to n less than or equal to N}, where g is a primitive root modulo an odd prime p. and let f(g)(m, H) denote the n umber of elements of V-v(g) that lie in the interval (m, m + H], where 1 le ss than or equal to m less than or equal to p. H. Montgomery calculated the asymptotic size of the second moment of f(g)(m, H) about its mean for a ce rtain range of the parameters N and H and asked to what extent this range c ould be increased if one were to average over all the primitive roots (mod p). We address this question as well as the related one of averaging over t he prime p. (C) 2001 Academic Press.