Simultaneous Primitive Roots over Finite Rings

This note investigates the average density of prime numbers $p\in [x,2x]$ with respect to a random simultaneous primitive root $g\le p^{1\mathrm{/}2+\epsilon }$ over the finite rings $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ and $\mathbb{Z}\mathrm{/}{p}^2\mathbb{Z}$ as $x\to \mathrm{\infty }$.