On primitive roots for Carlitz modules

AbstractLet A be the polynomial ring over a finite field. We prove that for every element a of a global A-field of finite A-characteristic the set of places P for which a is a primitive root under the Carlitz action possesses a Dirichlet density. We also give a criterion for this density to be positive. This is an analogue of Bilharz’ version of the primitive roots conjecture of Artin, with Gm replaced by the Carlitz module