On the Distribution of Small Powers of a Primitive Root

AbstractLet 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

A hybrid Euler-Hadamard product for the Riemann zeta function

We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theor