On an identity of Eckford Cohen

Abstract:

We characterize all multiplicative arithmetical functions ${f_k}(r)$ such that an identity of the form \[ \sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,} \] holds for all n, where ${q_k}(n)$ is the characteristic function of the set of k-free integers and ${c_k}(n,r)$ is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking ${f_k}(r) = \mu (r)/{J_k}(r)$ and $g(k) = \zeta (k)$.