Artin's conjecture on average for composite moduli
Carmichael extended the concept of primitive root by saying that a primitive \(\lambda\)-root \(\bmod n\) is a number with the maximal order \(\bmod n\). The author continues his work [Acta Arith. 86, No. 2, 113-132 (1998; Zbl 0917.11001)] on the number \(N_a(x)\) of integers \(n\leq x\) for which \(a\) is a primitive \(\lambda\)-root \(\bmod n\). For the corresponding number \(P_a(x)\) of primes \(p\leq x\) for which \(a\) is a primitive root, an asymptotic formula was deduced by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)] from a rather far-reaching extension of the Riemann Hypothesis. For the average \(\sum_{a<N}P_a(x)\) unconditional asymptotic formulae were obtained by \textit{D. Goldfeld} [Mathematika, London 15, 223--226 (1968; Zbl 0186.08401)] and by a rather different method by \textit{P. J. Stephens} [ Mathematika, London 16, 178--188 (1969; Zbl 0186.08402)]. One may enquire whether similar results hold for \(N_a(x)\), and in particular whether \(N_a(x)/x\) tends to a limit as \(x \to \infty\). The author remarks that this is doubtful, because he shows that the upper and lower limits of \(\sum_{1\leq a\leq x}N_a(x)/x^2\) are non-zero and of opposite sign.
scientific article; zbMATH DE number 1533159
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