Abstract :
Let a be an integer ≠−1 and not a square. Let Pa(x) be the number of primes up to x for which a is a primitive root. Goldfeld and Stephens proved that the average value of Pa(x) is about a constant multiple of x/ln x. Carmichael extended the definition of primitive roots to that of primitive λ-roots for composite moduli n, which are integers with the maximal order modulo n. Let Na(x) be the number of natural numbers up to x for which a is a primitive λ-root. In this paper we will prove that the average value of Na(x) oscillates. That is, imagex→∞ 1/x2 ∑1less-than-or-equals, slantaless-than-or-equals, slantx Na(x)>0 and imagex→∞ 1/x2 ∑1less-than-or-equals, slantaless-than-or-equals, slantx Na(x)=0.
