A non-abelian large sieve and Artin's primitive root conjecture

Abstract

A well-known conjecture of Artin states that if $a$ is an integer not equal to $0,\pm 1$ or a perfect square, then there exist infinitely many primes $p$ such that $a$ is a primitive root $(\text{ mod }p)$. In this article, we study a generalization of the classical (abelian) large sieve inequality in non-abelian settings. Assuming the non-abelian large sieve inequality, we provide a proof of Artin's primitive root conjecture. Further, using duality techniques, we derive unconditional results towards the conjectured non-abelian large sieve inequality.