Skip to main content

Research Repository

Advanced Search

Squarefrees are Gaussian in short intervals

Gorodetsky, Ofir; Mangerel, Alexander; Rodgers, Brad

Squarefrees are Gaussian in short intervals Thumbnail


Ofir Gorodetsky

Brad Rodgers


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 ! 1 and H D Xo.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.


Gorodetsky, O., Mangerel, A., & Rodgers, B. (2023). Squarefrees are Gaussian in short intervals. Journal für die reine und angewandte Mathematik, 2023(795), 1-44.

Journal Article Type Article
Online Publication Date Dec 6, 2022
Publication Date 2023
Deposit Date Dec 23, 2022
Publicly Available Date Mar 29, 2023
Journal Journal für die reine und angewandte Mathematik (Crelles Journal)
Print ISSN 0075-4102
Electronic ISSN 1435-5345
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 2023
Issue 795
Pages 1-44


You might also like

Downloadable Citations