On the distribution of square-full and cube-full integers

Let N-r(x) be the number of r-full integers less than or equal to x and let Delta(r)(x) be the error term in the asymptotic formula for N-r(x). Under Riemann's hypothesis, we prove the estimates Delta(2)(x) much less than x(1/7+epsilon), Delta(3)(x) much less than x(97/804+epsilon) (For All epsilon > 0), which improve those of Cao and Nowak. We also investigate the distribution of r-full and I-free numbers in short intervals (r = 2, 3). Our results sharpen Kratzel's estimates.