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.

Corrected and improved arguments. Accepted by Illinois Journal of Mathematics