Squarefrees are Gaussian in short intervals

Abstract

Abstract We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H → ∞ {H\to\infty} and H = X o ⁢ ( 1 ) {H=X^{o(1)}} . This answers a question posed by R. R. Hall in 1989. More generally, we prove a variant of Donsker’s theorem, showing that these counts scale to a fractional Brownian motion with Hurst parameter 1 / 4 {1/4} . In fact, we are able to prove these results hold in general for collections of B-free integers as long as the sieving set B satisfies a very mild regularity property, for Hurst parameter varying with the set B.