A note on the partial sum of Apostol's Möbius function

T. M. Apostol introduced a certain Möbius function \(\mu_{k}(\cdot)\) of order k, where \(k\geq 2\) is a fixed integer. Let k=1, then \(\mu_{1}(\cdot)\) coincides with the Möbius function \(\mu(\cdot)\), in the usual sense. For any fixed \(k\geq 2\), he proved the asymptotic formula \(\sum_{n\leq x}\mu_{k}(n)=A_{k}x+O_{k}(x^{1/k}\log x)\) as \(x\to\infty\), where \(A_{k}\) is a positive constant. Later, under the Riemann Hypothesis, D. Suryanarayana showed the O-term is \(O_{k}\bigl(x^{\frac{4k}{4k^{2}+1}}\exp\bigl(D\frac{\log x}{\log\log x}\bigr)\!\bigr)\) with some positive constant D. In this paper, without using any unproved hypothesis we shall prove that the O-term obtained by Apostol can be improved to \(O_{k}\bigl(x^{1/k}\exp\bigl(-D_{k}\frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\bigr)\!\bigr)\) with some positive constant \(D_{k}\).