Some asymptotic formulae involving Cohen-Ramanujan expansions

Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):={\sum }_{\begin{array}{c}h=1\\ (h,r)=1\end{array}}^r{e}^{\frac{2\pi inh}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan expansion $f(n):=\sum _r\widehat{f}(r)c_r^s(n)$ if the series on the right hand side converges for suitable complex numbers $\widehat{f}(r)$. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for the sum $\sum _{\begin{array}{c}n\le N\end{array}}f(n)g(n+h)$ where $h$ is a fixed non negative integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.