Sums of logarithmic weights involving r-full numbers

Abstract

Let (n, q) denote the greatest common divisor of positive integers n and q, and let \(f_{r}\) denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full (\(r=2\)) and cube-full numbers (\(r=3\)), which is \(\sum _{n\le y}\sum _{q\le x}\sum _{d|(n,q)}df_{r}\left( \frac{q}{d}\right) \log \frac{x}{q}\) with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with \(r=2\) under the Riemann Hypothesis.