Some average results connected with reductions of groups of rational numbers

Abstract

Let 􀀀 _ Q_ be a _nitely generated subgroup and let p be a prime number such that the reduction group 􀀀p is a well de_ned subgroup of the multiplicative group F_p. Firstly, given that 􀀀 _ Q_, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the average over prime numbers, powers of the order of the reduction group modulo p. The problem was previously considered by Pomerance and Kulberg for the rank 1 case. When 􀀀 contains only positive numbers, we are also able to give an explicit expression for the involved density in terms of an Euler product. The _rst part is concluded with some numerical computations. In the second part, for any m 2 N we prove an asymptotic formula for the average of the number of primes p _ x for which the index [F_p : 􀀀p] = m. The average is performed over all _nitely generated subgroups 􀀀 = ha1; : : : ; ari _ Q_, with ai 2 Z and ai _ Ti with a range of uniformity: Ti > exp(4(log x log log x) 1 2 ) for every i = 1; : : : ; r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar range of uniformity. The case of rank 1 and m = 1 corresponds to the classical Artin conjecture for primitive roots and has already been considered by Stephens in 1969.