Antonella Perucca

Artin's conjecture on primitive roots

If a is a non-zero integer and p varies among the prime numbers, we say that a is a primitive root modulo p if every non-zero residue class modulo p is a power of a mod p. In 1927, Artin predicted the density of the primes p for which a is a primitive root. Artin's conjecture has been proven by Hooley in 1967 conditionally under GRH. In 1975, Cooke and Weinberger generalized Hooley's result from Q to number fields. Beyond illustrating the classical results, we will present the Master theorem on Artin type problems (j.w. Järviniemi), a concrete understanding of when the Artin density is zero (j.w. Cherubini), uniform bounds for the ratio between the Artin density and a suitable Artin constant (j.w. Shparlinski) and a closed formula for the Artin density over any number field.