Squarefrees are Gaussian in short intervals
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909346922758144 |
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| author | Gorodetsky, Ofir Mangerel, Alexander P. Rodgers, Brad |
| author_facet | Gorodetsky, Ofir Mangerel, Alexander P. Rodgers, Brad |
| contents | 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 $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_12234 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Squarefrees are Gaussian in short intervals Gorodetsky, Ofir Mangerel, Alexander P. Rodgers, Brad Number Theory Probability 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 $B$. |
| title | Squarefrees are Gaussian in short intervals |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2112.12234 |