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 $0<\mathit{\theta }<1∕2$, almost all short intervals $(x,x+x^{1∕2+\mathit{\theta }}]$ contain about $\frac{\mathit{\zeta }(3∕2)}{2\mathit{\zeta }(3)}{x}^{\mathit{\theta }}$ squarefull numbers.