On extending Artin's conjecture to composite moduli in function fields

Abstract

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A=Fq[t] be the ring of polynomials over the finite field Fq and 0≠a∈A. Let C be the A-Carlitz module. Let a be a fixed element in A. For n∈A, C(A/nA) is a finite A-module. The set of all annihilators of C(A/nA) is an ideal and generated by a monic polynomial, denoted by λ(n). Similarly, The set of all annihilators of the submodule of C(A/nA) generated by a is an ideal and let la(n) be its monic generator. We say that a is a primitive root of n, if λ(n)=la(n). DefineNa(x):=|{n∈A|deg⁡n=x,nis monic,ais a primitiveroot of n}|We prove that for a given non-constant a∈A, a∉E, an exceptional set, there exists an unbounded set V of integers such thatliminfx∈VNa(x)/qx=0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields.