On the average number of power residues modulo a composite number

We study the behaviour of the quantities $a_{n}(q)$ and $b_{n}(q)$, that is, the number of $n$th power residues in the reduced and complete residue systems modulo a composite number $q$, respectively, where $n\geqslant2$ is an arbitrary fixed number. In particular, we prove asymptotic formulae for the sum functions $A_{n}(x)$ and $B_{n}(x)$ of these quantities.