Permutations with Orders Coprime to a Given Integer - Bamberg, John; Glasby, S. P.; Harper, Scott; Praeger, Cheryl E. (The Electronic Journal of Combinatorics, 2020)
Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group $\mathrm{Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geq m$,\[C(m) \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1} \leq \rho(n,m) \leq \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1}\]where $\phi$ is Euler's totient function.