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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2606.01384 |
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| _version_ | 1866917552754524160 |
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| author | Terada, Reo |
| author_facet | Terada, Reo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01384 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sums of Apostol's Möbius functions of order $k$ Terada, Reo Number Theory 11N37 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. |
| title | Sums of Apostol's Möbius functions of order $k$ |
| topic | Number Theory 11N37 |
| url | https://arxiv.org/abs/2606.01384 |