Two identities involving Cohen-Ramanujan expansions

Abstract

An arithmetical function $f$ is said to admit a \emph{Cohen-Ramanujan expansion} $f(n):=\sum _r\hat{f}(r)c_r^s(n)$, if the series on the right hand side converges for suitable complex numbers $\hat{f}(r)$. Here $c_r^s(n)$ denotes the Cohen-Ramanujan sum defined by E. Cohen. We deduce here a Cohen-Ramanujan expansion for the Jordan totient function $J_k(n)$. Further, we give an an asymptotic formula for the sum $\sum _{n\le N}\frac{J_a(n)}{n^a}\frac{J_b(n+h)}{(n+h{)}^b}$ using the expansion we derive.