A lower bound to the error term of a function related to \(k\)th order free integers (mod \(M\)) (Q1346441)

For fixed natural numbers \(k\), \(m\) denote by \(g_ k (m)\) the number of integers \(t\bmod m\) satisfying \(t^{n+k} \equiv t^ k\bmod m\) for some natural number \(n\). Then \(g_ k\) is multiplicative and the asymptotic formula \[ \sum_{m\leq x} g_ k (m^ k)= {\textstyle {c \over {k+1}}} x^{k+1}+ E(x) \] holds with a certain constant \(c\). In the present paper the author shows \(E(x)= \Omega (x^ k \log \log \log x)\).