Roots of unity and nullity modulo $n$

For a fixed positive integer $\ell$, we consider the function of $n$ that counts the number of elements of order $\ell$ in $\mathbb{Z}_n^*$. We show that the average growth rate of this function is $C_\ell(\log n)^{d(\ell)-1}$ for an explicitly given constant $C_\ell$, where $d(\ell)$ is the number of divisors of $\ell$. From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo $n$ of order $\ell$ is $(d(\ell)-1)C_\ell (\log n)^{d(\ell)-2}$ for $\ell\ge2$. We also consider the number of elements of $\mathbb{Z}_n$ whose $\ell$th power equals 0, showing that its average growth rate is $D_\ell(\log n)^{\ell-1}$ for another explicit constant $D_\ell$. Two techniques for evaluating sums of multiplicative functions, the Wirsing-Odoni and Selberg-Delange methods, are illustrated by the proofs of these results.