The Artin–carmichael Primitive Root Problem on Average

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (Z/nZ). For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (Z/nZ). Let R(n) denote the number of elements of (Z/nZ) with order λ(n). It is natural to compare Na(x) with ∑ n≤x R(n)/n. In this paper we show that the average of Na(x) for 1≤ a ≤ y is indeed asymptotic to this sum, provided y ≥ exp((2+ ε)(log x log log x)1/2), thus improving a theorem of the first author who had this for y ≥ exp((log x)3/4). The result is to be compared with a similar theorem of Stephens who considered the case of prime numbers n. §