Squarefrees are Gaussian in short intervals

Gorodetsky, Ofir; Mangerel, Alexander; Rodgers, Brad

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 ! 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.

Citation

Gorodetsky, O., Mangerel, A., & Rodgers, B. (2023). Squarefrees are Gaussian in short intervals. Journal für die reine und angewandte Mathematik, 2023(795), 1-44. https://doi.org/10.1515/crelle-2022-0066

• 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
• DOI: https://doi.org/10.1515/crelle-2022-0066