Sums of logarithmic weights involving r-full numbers.

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 ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f r q d log 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. [ABSTRACT FROM AUTHOR]