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∑n⩽xf([x/n])[x/n]k as x→∞, where [⋅] 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 Bordellès-Dai-Heyman-Pan-Shparlinski.