Permutations with orders coprime to a given integer

Let $m$ be a positive integer and let $\rho (m,n)$ be the proportion of permutations of the symmetric group $\mathrm{S}\mathrm{y}\mathrm{m}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$where$\kappa_m$isacomplicated(unbounded)functionof$m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geqslant m$, $$C(m){\left(\frac{n}{m}\right)}^{\frac{\varphi (m)}{m}-1}⩽\rho (n,m)⩽{\left(\frac{n}{m}\right)}^{\frac{\varphi (m)}{m}-1}$$ where $\varphi$ is Euler’s totient function.