Averages of Ramanujan sums: note on two papers by E. Alkan (Q2254856)

The author proves identities of weighted averages \[ {1\over k^{r+1}}\,\sum^k_{j=1} \omega(j)\,c_k(j) \] of Ramanujan sums \(c_k(j)\). The case \(\omega(j)= j^r\) was proved by \textit{E. Alkan} [ibid. 29, No. 1--3, 385--408 (2012; Zbl 1297.11092)]. The author gives a shorter proof of Alkan's result and generalizes to other weights like \(\omega(j)=\log j\), an arithmetic function of \(\text{gcd}(j,k)\), Bernoulli numbers and polynomials, and \(\log\Gamma({j\over k})\). He gives a multivariable generalization.