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On a sum involving the Möbius function

Volume 169 / 2015

I. Kiuchi, M. Minamide, Y. Tanigawa Acta Arithmetica 169 (2015), 149-168 MSC: 11N37, 11L03. DOI: 10.4064/aa169-2-3

Abstract

Let $c_{q}(n)$ be the Ramanujan sum, i.e. $c_{q}(n)=\sum_{d|(q,n)}d \mu(q/d)$, where $\mu$ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for $\sum_{n\leq y}(\sum_{q\leq x}c_{q}(n))^{k}$ ($k=1,2$) are obtained. As an analogous problem, we evaluate $\sum_{n\leq y}(\sum_{n\leq x}\widehat{c}_{q}(n))^{k}$ ($k=1,2$), where $\widehat{c}_{q}(n):=\sum_{d|(q,n)}d|\mu(q/d)|$.

Authors

  • I. KiuchiDepartment of Mathematical Sciences
    Faculty of Science
    Yamaguchi University
    Yoshida 1677-1
    Yamaguchi 753-8512, Japan
    e-mail
  • M. MinamideDepartment of Mathematical Sciences
    Faculty of Science
    Yamaguchi University
    Yoshida 1677-1
    Yamaguchi 753-8512, Japan
    e-mail
  • Y. TanigawaGraduate School of Mathematics
    Nagoya University
    Nagoya 464-8602, Japan
    e-mail

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