Artin's primitive root conjecture: classically and over Fq[T]

In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g is a primitive root mod p. By observing numerical deviations from Artin's originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor; leading to a modified conjecture that was eventually proved correct by Hooley (1967), under the assumption of the Generalized Riemann Hypothesis (GRH). In this talk we discuss several variants of Artin's conjecture: namely an "Artin Twin Primes Conjecture", as well as an appropriate analogue of Artin's primitive root conjecture for algebraic function fields.