On Ramanujan expansions with additive coefficients

The Ramanujan series attached to a complex-valued arithmetic function $\widehat g$ in a fixed integer $a$ is the series $\sum_n\widehat g(n)c_n(a)$, where $c_n(a)$ is the so-called Ramanujan sum. Assuming that $\widehat g$ is additive or, more generally, a product of a multiplicative function with an additive one, we study the relationships between the Ramanujan series attached to $\widehat g$ in a positive integer $a$ and its subseries obtained by taking the terms with $n$ coprime to a fixed integer $d\ge 2$.