Almost 100 years of Artin’s conjecture on primitive roots (online)

If $a$ is an integer and $p$ a prime number, we say that a is a primitive root modulo $p$ if the powers of $a$ are representants for all the non-zero residue classes modulo $p$. In 1927, Artin conjectured that, provided that $a$ is neither a square nor $-1$, the set of primes $p$ such that a is a primitive root modulo $p$ has a positive natural density (with a precise expression for the density). In 1967, Hooley was able to prove (a slightly corrected version of) Artin’s conjecture assuming GRH. The correction is related to an interesting phenomenon now called entanglement. Up to date, there is no integer a that is known (unconditionally) to be a primitive root modulo $p$ for infinitely many primes $p$. Beyond the classical results, we will present new developments. Assuming GRH, with Järviniemi and Sgobba we have obtained very general results on Artin-type problems, and with Shparlinski we have discovered uniform bounds on Artin’s density. We will also address the computational problem of determining a primitive root modulo p by testing candidates.