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.
Citation
Pieter Moree. "On primes $p$ for which $d$ divides $\mathrm{ord}_p (g)$." Funct. Approx. Comment. Math. 33 85 - 95, 2005. https://doi.org/10.7169/facm/1538186603
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