Higher rank generalizations of Fomenkoʼs conjecture

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 Fp⁎. Assuming the generalized Riemann hypothesis and Hypothesis A of Hooley, Fomenko proved in 2004∑p⩽xlog(ia(p))=cali(x)+O(xloglogx(logx)2), where ca is a constant dependent on a, and where li(x) is the logarithmic integral. We prove a higher rank version of this result without using Hypothesis A of Hooley. More precisely, let {a1,a2,…,ar}⊂Q⁎ be a multiplicatively independent set of integers. Let Γ=〈a1,a2,…,ar〉 be the group generated by a1,a2,…,ar in Q⁎. For primes p, define iΓ(p) to be [(Z/pZ)⁎:Γmodp], where Γmodp is the group generated by a1,a2,…,ar inside Fp⁎. We show that, for r⩾2, there is a positive constant cΓ>0 such that∑p⩽xlogiΓ(p)=cΓli(x)+O(xθ), where θ<1.