Squarefrees are Gaussian in short intervals

TL;DR

This paper proves that counts of squarefree integers in short intervals follow a Gaussian distribution and scale to fractional Brownian motion, answering a long-standing question and extending to B-free integers under mild conditions.

Contribution

It establishes Gaussian distribution and fractional Brownian motion scaling for squarefree and B-free integers in short intervals, generalizing previous results.

Findings

Counts of squarefree integers tend to a Gaussian distribution in short intervals.

These counts scale to a fractional Brownian motion with Hurst parameter 1/4.

Results hold for B-free integers under mild regularity conditions.

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 $H\rightarrow\infty$ 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 $B$.