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

Abstract

LetN r (x) be the number ofr-full integers ≤x and let Δ r (x) be the error term in the asymptotic formula forN r (x). Under Riemann's hypothesis, we prove the estimates Δ2(x)≪x1/7+e, Δ3(x)≪x97/804+e(∀e>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Kratzel's estimates.