On sums of sums involving the Liouville function (Q6612916)

Consider the Liouville function \(\lambda\) defined by \(\lambda(n)=(-1)^{\Omega(n)}\), where \(\Omega(n)\) represents the number of prime divisors of \(n\), counted with multiplicity. The function\N\[\Ns_q(n)= \sum_{d\mid (q,n)} d\ \lambda(q/d)\N\]\Nis an analogue of the Ramanujan sum\N\[\Nc_q(n)= \sum_{d\mid (q,n)} d\ \mu(q/d),\N\]\Nwhere \(\mu\) denotes the Möbius function.\N\NIn the present paper the author derives asymptotic formulas for the sums\N\[\N\sum_{n\le y} \left(\sum_{q\le x} s_q(n)\right)^k\N\]\Nin the cases \(k=1\) and \(k=2\). Similar results concerning the Ramanujan sums \(c_q(n)\) have been investigated by \textit{T. H. Chan} and \textit{A. V. Kumchev} [Acta Arith. 152, No. 1, 1--10 (2012; Zbl 1294.11162)].