Primes in arithmetic progressions and nonprimitive roots

Abstract

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z}{)}^{\ast },$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.

7 pages, changed title, to appear in the Bulletin of the Australian Mathematical Society, deals with a side problem that came up in arXiv:1809.08431 (to appear in Journal of Number Theory)