Ramanujan sums are nearly orthogonal to powers

Abstract

Sign of averages of Ramanujan sums is studied and it is shown that these averages have a curious tendency to be positive. This in turn gives that Ramanujan sums are nearly orthogonal to a family of vectors whose entries are powers of consecutive integers. Further applications are given to the limit points of these averages along semigroups of integers, the peak size of partial sums of Ramanujan sum and an optimization problem on weighted exponential sums supported on reduced residue systems. Exact evaluations of trigonometric sums having combinatorially significant coefficients and subject to divisibility constraints are obtained in terms of Bernoulli numbers.