Estimates of certain arithmetic sums related to the number of divisors (Q881127)

Let \(\tau_k(n)\) denote the number of ways in which \(n\) can be represented as a product of \(k\geq 1\) factors, so that \(\tau_1(n) \equiv 1\) and \(\tau_2(n) \equiv \tau(n)\) = the number of divisors of \(n\). The author proves that \[ \sum_{n\leq X}\tau_{k_1}(n)\cdots\tau_{k_\ell}(n) \leq {X\over(k_1\cdots k_\ell - 1)!} (\log X + k_1\cdots k_\ell - 1)^{k_1\cdots k_\ell - 1} \leqno(1) \] holds for natural numbers \(\ell\geq 1, k_1 \geq 2,\ldots, k_\ell\geq 2\) and \(X\geq 1\). The proof of (1), which refines previous results of \textit{K. K. Mardzhanishvili} [Dokl. Akad. Nauk SSSR 22, 387--389 (1939; Zbl 0021.20802)] and the author [Chebyshevskiĭ\ Sb. 6, No. 1(13), 146--156 (2005; Zbl 1142.11068)], is elementary. It is based on the inequality \[ \tau_k(n)\tau_\ell(n) \leq \tau_{k\ell}(n)\qquad(k\geq2, \ell\geq2, n\geq 1), \] which easily follows from the fact that \(\tau_k(n)\) is a multiplicative function of \(n\).