Roots of unity and nullity modulo $n$

Abstract:

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 $\ell \ge 2$. 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.