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 E. Alkan in [J. Number Theory, 140:147–168 (2014)]. Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by E. Cohen in [Duke Math. J., 16(2):85–90 (1949)]. We also prove that the weighted average $\frac{1}{k^{s(r+1)}}{\sum }_{j=1}^{k^s}{j}^r{c}_k^{(s)}(j)$ remains positve for all $r\ge 1$. Further, we give a lower bound for $\underset{N}{\max }\mid {\sum }_{j=1}^{N^s}{c}_k^{(s)}(j)\mid$.