Sums of Apostol's Möbius functions of order $k$

Abstract

In 1970, T. M. Apostol introduced the Möbius function $μ_{k}$ of order $k$ for all positive integer $k$, as a generalization of the Möbius function $μ= μ_{1}$. For any integer $k \ge 2$, he proved $\sum_{n \le x} μ_{k}(n) = A_{k} x + O_{k}(x^{1/k} \log x)$ where $A_{k}$ is a positive constant. In 2001, A. Bege conjectured both the conditional and unconditional estimates for the sum $\sum_{n \le x, (n, q) = 1}μ_{k}(n)$ for any positive integer $q$. In this paper, we give affirmative solutions to the conditional version of Bege's conjecture completely and the unconditional one partially. We also give a mean square estimate for the error term.