In 1957, M. A. Subhankulov established the following identity
where ; μ is the Môbius function and J
2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(n
r) = ∑d| (n, r)
dμ(r/d), we rewrite the above identity using C(n, r).
In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.