Artin's primitive root conjecture for function fields revisited
Artin's primitive root conjecture claims that for each integer \(g > 0\), either \(g\) is a perfect square or there exist infinitely many primes \(p\) such that \(g\) is a primitive root modulo \(p\). The corresponding property for function fields was proved by \textit{H. Bilharz} [Math. Ann. 114, 476--492 (1937; Zbl 0016.34301)] assuming the Riemann hypothesis for function fields, which was later proved by André Weil. In this article the authors prove Bilharz's theorem without using the Riemann hypothesis for function fields; instead they apply an idea used by Hadamard and de la Vallée Poussin for proving the prime number theorem: it is sufficient to obtain a sufficiently large zero-free region of the congruence zeta functions. The proof follows the strategy already used by \textit{E. Jensen} and \textit{M. R. Murty} [in: Number theory. Basel: Birkhäuser. 167--181 (2000; Zbl 0987.11074)].
scientific article; zbMATH DE number 7312701
scientific article