On a sum involving the Möbius function
Volume 169 / 2015
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 $\widehat{c}_{q}(n):=\sum_{d|(q,n)}d|\mu(q/d)|$.
