A NOTE ON POWERFUL NUMBERS IN SHORT INTERVALS
A NOTE ON POWERFUL NUMBERS IN SHORT INTERVALS
Abstract We investigate uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$ . We obtain unconditional upper bounds $O({y}/{\log y})$ and $O(\kern1.3pt y^{11/12})$ for all powerful numbers and $y^{1/2}$ -smooth powerful numbers, respectively. Conditional on the $abc$ -conjecture, we prove the bound $O({y}/{\log …