On sums of number-theoretic functions (Q2365007)

\textit{R. L. Duncan} [Proc. Am. Math. Soc. 25, 191-192 (1970; Zbl 0216.03701)] proved by elementary methods \[ \sum_{2\leq n\leq x}{1\over{\omega(n)}}\ll x/\log\log x\quad\text{ and }\quad \sum_{2\leq n\leq x}{{\Omega(n)} \over {\omega(n)}}=x+ O(x/\log\log x). \] Using strong analytic methods De Koninck, Ivić et al. sharpened the results of Duncan and got several result on reciprocals of additive functions [cf. \textit{J.-M. De Koninck} and \textit{A. Ivić}, Topics in arithmetical functions, North-Holland Mathematics Studies 43, Amsterdam (1980; Zbl 0442.10032) for a collection]. We use a result of \textit{H. Nakaya} [Sci. Rep. Kanazawa Univ. 37, 23-47 (1992; Zbl 0774.11056)] to generalize the results of De Koninck and Ivić to arithmetic progressions and give a sharp asymptotic formula for sums of the type \[ \mathop{{\sum}'}_{\substack{ n\leq x\\n\equiv l\pmod k}} \chi(n)f_1(n)^{\nu_1}\cdot \cdots\cdot f_r(n)^{\nu_r}, \] where \(\chi\) is a suitable multiplicative function, \(f_1,\dots,f_r\) are ``small'' additive, prime-independent arithmetic functions, \(\nu_1,\dots,\nu_r\in\mathbb{Z}\) are arbitrary and \(k,l\) are coprime. The summation is over those \(n\) not exceeding \(x\) for which \(f_1(n)\cdot \cdots\cdot f_r(n)\neq 0\). The proof is based on a method of De Koninck and Ivić, described in (loc. cit.) and outlined in [\textit{A. Ivić}, Publ. Inst. Math., Nouv. Ser. 41(55), 31-41 (1987; Zbl 0648.10028)].