Geodesic
Sums of products of generalized Ramanujan sums
Journal of integer sequences, Tome 19 (2016) no. 2
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.
Classification : 11A25, 11B68
Keywords: arithmetical function, Ramanujan's sum, greatest common divisor 
@article{JIS_2016__19_2_a5,
     author = {Ikeda,  Soichi and Kiuchi,  Isao and Matsuoka,  Kaneaki},
     title = {Sums of products of generalized {Ramanujan} sums},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {2},
     zbl = {1415.11013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_2_a5/}
}
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Ikeda,  Soichi; Kiuchi,  Isao; Matsuoka,  Kaneaki. Sums of products of generalized Ramanujan sums. Journal of integer sequences, Tome 19 (2016) no. 2. http://geodesic.mathdoc.fr/item/JIS_2016__19_2_a5/