Artin's conjecture for primitive roots
This paper is a nice expository article discussing work by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)], \textit{R. Gupta} and the author [Invent. Math. 78, 127-130 (1984; Zbl 0549.10037)], and by the reviewer [Q. J. Math., Oxf. II. Ser. 37, 27-38 (1986; Zbl 0586.10025)]. These authors are concerned with Artin's conjecture that if k is any integer other than -1 or a square, then there are infinitely many primes for which k is a primitive root. Hooley proved the conjecture, and even an asymptotic formula for the density of such primes, subject to the Riemann hypothesis for certain Dedekind zeta- functions. The other authors proved unconditional results, showing that the conjecture does at least hold for most values of k, in an appropriate sense. In this article the underlying ideas behind the various proofs are explained, and the corresonding question for elliptic curves is discussed.
scientific article; zbMATH DE number 4071031
scientific article