The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

Abstract

Let [Formula: see text] be a finitely generated subgroup of [Formula: see text]. Let [Formula: see text] be a prime number for which the reduction group [Formula: see text] is a well-defined subgroup of the multiplicative group [Formula: see text], and denote the order of [Formula: see text] by [Formula: see text]. Assuming the Generalized Riemann Hypothesis, we study the average of [Formula: see text] and the average of the powers of [Formula: see text] as [Formula: see text] ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when [Formula: see text] contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.