Cohen-Ramanujan sum, denoted by \(c_r^s(n)\) , is an exponential sum similar to the Ramanujan sum \(c_r(n):=\sum \limits _{\begin{array}{c} h=1\\ {(h,r)=1} \end{array}}^{r}e^{\frac{2\pi i n h}{r}}\) . An arithmetical function f is said to admit a Cohen-Ramanujan expansion \( f(n):=\sum \limits _{r}\widehat{f}(r)c_r^s(n)\) 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 \(\sum \limits _{\begin{array}{c} n\le N \end{array}}f(n)g(n+h)\) where h is a nonnegative integer. We further show that a Cohen-Ramanujan expansion for \(f_h(n):=f(n+h)\) exists whenever such an expansion exists for f.
