On arithmetical functions related to the Ramanujan sum (Q1416132)

Let \(m\) and \(n\) be natural numbers, \(f\) be an arbitrary arithmetical function and \(\mu\) be the Möbius function. Let \[ S_f(m,n)=\sum_{d|(m,n)} d\mu\left(\frac{m}{d}\right)f\left(\frac{n}{d}\right). \] It is shown that the function \(S_f\) has a lot of the properties of the Ramanujan sum. When \(f=\mu\) a Dirichlet series is obtained. The following relation between \(S_f\), \(\mu\) and Dedekind functions is proved: \[ S_\mu(m,n)=\mu\frac{[m,n]}{(m,n)} \frac{J(mn)}{J\left(\frac{[m,n]}{(m,n)}\right)} \frac{1}{(m,n)}, \] where for \(n=\prod^k_{i=1}\) \(p^{\alpha_i}_i\) (\(p_1,\dots,p_k\) are prime numbers, \(\alpha_1,\dots,\alpha_k\) are natural numbers) the Dedekind function is \(J(n)=n\prod^k_{i=1}\left(1+\frac{1}{p}\right)\).