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 \(\frac{1}{k^{s(r+1)}}\sum \limits _{j=1}^{k^s}j^rc_k^{(s)}(j)\) $\frac{1}{k^{s(r+1)}}\sum _{j=1}^{k^s}{j}^r{c}_k^{(s)}(j)$ remains positive for all \(r\ge 1\) $r\ge 1$ . Further, we give a lower bound for \(\max \limits _{N}\left| \sum \limits _{j=1}^{N^s}c_k^{(s)}(j) \right| \) $\underset{N}{max}\left|\sum _{j=1}^{N^s},c_k^{(s)},(j)\right|$ .