On primitive roots for Carlitz modules

Let 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 B 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 G(m) replaced by the Carlitz module. (C) 2003 Elsevier Science (USA). All rights reserved.