Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.7

Sums of Products of Generalized Ramanujan Sums


Soichi Ikeda
Department of Mathematics
Shibaura Institute of Technology
307 Fukasaku, Minuma-ku, Saitama 337-8570
Japan

Isao Kiuchi
Department of Mathematical Sciences
Faculty of Science
Yamaguchi University Yoshida 1677-1 Yamaguchi 753-8512
Japan

Kaneaki Matsuoka
Graduate School of Mathematics
Nagoya University
Furocho Chikusaku Nagoya 464-8602
Japan

Abstract:

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 fi, gi and hi $ (i=1, \ldots, n),\ $ and derive formulas for several weighted averages with weights concerning completely multiplicative functions, completely additive functions, and others.

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(Concerned with sequences A018804 A051193 A056188 A159068.)

Received August 18 2015; revised versions received November 21 2015; March 17 2016. Published in Journal of Integer Sequences, March 18 2016.

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