A PROBLEM OF FOMENKO'S RELATED TO ARTIN'S CONJECTURE

Abstract

Let a be a natural number greater than 1. For each prime p, let ia(p) denote the index of the group generated by a in [Formula: see text]. Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of ia(p) is constant without using Conjecture A of Hooley. More precisely, we show upon GRH that for any α with 0 ≤ α < 1, there is a positive constant cα > 0 such that [Formula: see text] where π(x) is the number of primes p ≤ x. We also study related questions.