Asymptotic formula for the multiplicative function \(\frac{d(n)}{k^{\omega(n)}}\) (Q6585102)

Let \(k\geq 2\) be a fixed integer, \(d(n) := \sum_{d\mid n}1\) be the number of divisors function, \(\omega(n):=\sum_{p\mid n}1\) be the number of distinct prime divisors function, and define the arithmetic function \(D_{k,\omega}(n)\) by \(D_{k,\omega}(n) :=d(n)/k^{\omega (n)}\). In the paper under review, the author obtains two results concerning the mean value of \(D_{k,\omega}(n)\). The weaker result, which is based on Tulyaganov's theorem, asserts that for all \(x\geq 1\) large enough, one has\N\[\N\sum_{n\leq x}D_{k,\omega}(n) =C_k\,x(\log x)^{2/k-1}+O(x(\log x)^{-1}(\log \log x)^{4/k}),\N\]\Nwhere\N\[\NC_k=\frac{1}{\Gamma (2/k)}\prod\limits_{p}\Bigl(1-\frac{1}{p}\Bigr) ^{2/k} \Bigl(1+\frac{2p-1}{kp(p-1) ^{2}}\Bigr).\N\]\NThen, the author improves on the \(O\)-term by replacing it by \(O_{k}(x(\log x)^{2/k-2})\).