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 1 k s ( r + 1 ) ∑ j = 1 k s j r c k ( s ) ( j ) remains positive for all r ≥ 1 . Further, we give a lower bound for max N ∑ j = 1 N s c k ( s ) ( j ) .