On the number of permutations of special form

It is shown that the number of permutations $a$ for which the equation $x^k=a$, where $a\in S_n$ ($S_n$ is the symmetric group of degree $n$) and $k<1$ is a fixed natural number, has at least one solution $x\in S_n$ is asymptotically equal to
$$ C(k)n^{\varphi(k)/k-1/2}\biggl(\frac ne\biggr)^n\quad\text{as}\quad n\to\infty, $$
where $C(k)$ is a constant depending only on $k$, and $\varphi(k)$ is the Euler function.
Bibliography: 4 titles.