On primes $p$ for which $d$ divides $\mathrm{ord}_p (g)$

Abstract

Let $N_{g}(d)$ be the set of primes $p$ such that the order of $g$ modulo $p$, $\mathrm{ord}_p (g)$, is divisible by a prescribed integer $d$. Wiertelak showed that this set has a natural density, $\delta_{g}(d)$, with $\delta_{g}(d) \in \mathbb{Q}_{>0}$. Let $N_{g}(d)(x)$ be the number of primes $p \leqslant x$ that are in $N_{g}(d)$. A simple identity for $N_{g}(d)(x)$ is established. It is used to derive a more compact expression for $\delta_{g}(d)$ than known hitherto.