A generalization of a result of K. R. Johnson (Q1823969)

In [Elem. Math. 38, 122-124 (1983; Zbl 0516.10002)] \textit{K. R. Johnson} evaluated the sum \(\sum_{d | n}| c_ k(n)|,\) where \(c_ k(n)\) is Ramanujan's trigonometric sum. In the present paper the author generalizes Johnson's result by evaluating the sum \(\sum_{d | n}| H_ k(d)|,\) where \(H_ k(n)\) is a generalization of Ramanujan's sum defined by \[ H_ k(n)=\sum_{d | (k,n)}\mu (k/d)h(d), \] \(\mu\) being the Möbius function and h being any multiplicative function. \textit{J. Chidambaraswamy} and \textit{P. V. Krishnaiah} [Int. J. Math. Math. Sci. 11, No.2, 351-354 (1988; Zbl 0652.10003)] have independently proved a similar type of evaluation for a more extensive class of generalizations of Ramanujan's sum.