A reciprocity theorem for the Cohen–Ramanujan sums and its application to Cohen–Ramanujan expansions

For an arithmetical function f, its Ramanujan expansion is a series expansion in the form [Formula: see text] where [Formula: see text] are complex numbers and [Formula: see text] is the Ramanujan sum. Here, we prove a reciprocity result on Cohen–Ramanujan sums [Formula: see text] to change the position of k and n in a twisted function and use it to prove that for certain arithmetical functions f, Cohen–Ramanujan series expansions in the form [Formula: see text] exist if and only if expansions in the form [Formula: see text] exist.