Variance of squarefull numbers in short intervals

Abstract

In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0<\theta<1/2$, almost all short intervals $(x, x + x^{1/2 + \theta}]$ contain about $\frac{\zeta(3/2)}{2 \zeta(3)} x^\theta$ squarefull numbers.