On near orthogonality of certain k-vectors involving generalized Ramanujan sums

The near orthgonality of certain k-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average \(\frac{1}{k^{s(r+1)}}\sum \limits _{j=1}^{k^s}j^rc_k^{(s)}(j)\) remains positive for all \(r\ge 1\). Further, we give a lower bound for \(\max \limits _{N}\left| \sum \limits _{j=1}^{N^s}c_k^{(s)}(j) \right| \).