Squarefrees are Gaussian in short intervals

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 $H\to \infty$ and $H=X^{o(1)}$. 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 $1/4$. 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 $B$.

34 pages, 6 figures. Accepted version. Minor changes (mostly typos and style)

References in corpus (4)

• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
• On invariant measures for $\mathcal{B}$-free systems
• Dynamics of $\mathcal{B}$-free sets: a view through the window
• On the variance of squarefree integers in short intervals and arithmetic progressions