In 1970, Tom M. Apostol
introduced a class of arithmetical functions μk(n) for all positive integral k, as a
generalization of the Möbius function μ(n) = μ1(n) and established the following
theorem: For k ≧ 2,Mk(x) = Σnx−μk(n) = Akx + O(x1∕klogx), where Ak is a positive
constant. In this paper we improve the above O-estimate to O(x4k∕(4k2+1)ω(x)) on
the assumption of the Riemann hypothesis, where ω(x) =exp{Alogx(loglogx)−1},A
being a positive absolute constant.