On an error term related to the Jordan totient function \(J_ k(n)\) (Q908957)

Let \(J_ k\) denote the Jordan totient function of order k and let \(E_ k(x)\) denote the error term in the asymptotic formula for the partial sums \(\sum_{n\leq x}J_ k(n)\). For \(k=1\), \(J_ k\) is the Euler totient function \(\phi\) and the investigation of \(E_ 1(x)\) has a long history, see \textit{D. Suryanarayana} [J. Indian Math. Soc., New. Ser. 42, 179-195 (1978; Zbl 0472.10045)] and \textit{H. L. Montgomery} [Proc. Indian Acad. Sci., Math. Sci. 97, 239-245 (1987; Zbl 0656.10042)]. The authors study the behaviour of \(E_ k(x)\) for \(k\geq 2\).