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.