Summary: We consider weighted averages for the products $ t_{k_1}^{(1)}(j)\cdots t_{k_n}^{(n)}(j) $of generalized Ramanujan sums $ t_{k_i}^{(i)}(j)=\sum_{d\vert\gcd(k_{i},j)}f_{i}(d)g_{i} ({k_i}/{d})h_{i}({j}/{d}) $with any arithmetical functions $f_{i}, g_{i}$ and $h_{i} (i=1, \ldots, n),\ $ and derive formulas for several weighted averages with weights concerning completely multiplicative functions, completely additive functions, and others.

11A25, 11B68

arithmetical function, Ramanujan's sum, greatest common divisor