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arXiv · 2606.01384

Sums of Apostol's M\"{o}bius functions of order $k$

Reo Terada
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Abstract

In 1970, T. M. Apostol introduced the M\"{o}bius function $\mu_{k}$ of order $k$ for all positive integer $k$, as a generalization of the M\"{o}bius function $\mu = \mu_{1}$. For any integer $k \ge 2$, he proved $\sum_{n \le x} \mu_{k}(n) = A_{k} x + O_{k}(x^{1/k} \log x)$ where $A_{k}$ is a positive constant. In 2001, A. Bege conjectured both the conditional and unconditional estimates for the sum $\sum_{n \le x, (n, q) = 1}\mu_{k}(n)$ 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.

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Reo Terada. 2026-05-31. Sums of Apostol's M\"{o}bius functions of order $k$. https://arxiv.org/abs/2606.01384

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