On a sum involving small arithmetic function and the integral part function

Let f be a "small" arithmetic function in the sense that f = g *1 and g(n) << n-j, where j is a fixed non-negative number. In this paper, we study the sum Sigma(n <= x) f([x/n])/[x/n](k) as x -> infinity, where [center dot] denotes the integral part function and k is a fixed non-negative number. Our results generalize the very recent work of Stucky, also combine and generalize the original two types of sums studied by Bordelles-Dai-Heyman-Pan-Shparlinski. (c) 2023 Elsevier Inc. All rights reserved.