On Ramanujan expansions with multiplicative coefficients

The Ramanujan sum cn(a) is related to the Möbius function μ, since cn(a) = μ(n) whenever a, n∈ N are such that gcd (n, a) = 1. An arithmetic function g admits a Ramanujan expansion in A⊆ N if g(a) = ∑ g^ (n) cn(a) for any a∈ A, where g^ is a suitable arithmetic function. Assuming that g^ is multiplicative, Coppola has recently proved that the convergence of the subseries ∑ gcd(n,a)=1g^ (n) μ(n) yields that of ∑ g^ (n) cn(a) by providing a factorization of the latter series in terms of the former. Such a factorization allows us to come across some explicit relationships between g and g^ without the benefit of the absolute convergence of the Ramanujan expansion. Among other things, we establish a recursive formula that relates the values attained at the prime powers by g and g^. Finally, we provide with a converse of Coppola’s results and a strengthening of ours in case g^ is a multiplicative function such that | g^ (p) | < 1 for any prime number p.