On Artin's conjecture over function fields
Let \(C\) be a nonsingular irreducible curve of degree \(d\) over a finite field \(\mathbb{F}_q\) and let \(K\), \(\overline K\) be the corresponding function fields over \(\mathbb{F}_q\) resp. over its algebraic closure. Let \(n\) be an integer and assume that \(r \in K\) is a rational function of degree \(m\) which for no divisor \(k\) of \(q^n - 1\) is the \(k\)-th power of a rational function in \(\overline K\). Then the number of prime divisors of \(C\) of degree \(n\) with respect to which \(r\) is a primitive root equals \(\varphi (q^n - 1)/n +\) error term, which in an explicit way depends on \(m,n,d,q\) and the number of distinct prime divisors of \(q^n - 1\). The novelty of this result lies in the error term, since asymptotics in this problem have been found already by \textit{H. Bilharz} [Math. Ann. 114, 476-492 (1937; Zbl 0016.34301)] who did not restrict himself to the case of rational \(r\). The proof is easily reduced to the count of primitive roots of \(\mathbb{F}_{q^n}\) having a special form and this is done using a bound for character sums obtained by \textit{G. I. Perelmuter} [Mat. Zametki, 5, 373-380 (1969; Zbl 0179.49903)] as a consequence of Riemann's hypothesis for function fields.
scientific article; zbMATH DE number 828728