On primitive roots for Carlitz modules. (Q1394910)

Let \(A\) be the polynomial ring over a finite field. Here the authors prove that for every element \(a\) of a global \(A\)-field of finite \(A\)-characteristic the set of places \(\mathcal P\) for which \(a\) is a primitive root under the Carlitz action possesses a Dirichlet density. They also give a criterion for this density to be positive. This is an analogue of Bilharz' version [\textit{H. Bilharz}, Math. Ann. 114, 476--492 (1937; Zbl 0016.34301 and JFM 63.0099.01)] of the primitive roots conjecture of Artin, with \(\mathbb G_m\) replaced by the Carlitz module.