### Abstract

In this survey, we outline several results on the distribution of B-free integers and explore a random process naturally associated to them. We show how, notwithstanding the rigid ergodic properties of this process (zero entropy, pure point spectrum, no weak mixing), it exhibits a central limit theorem resembling a theorem by Beck on the circle rotation by a quadratic surd. We explain the connection of the random process to the distribution of B-free integers in short intervals, with particular emphasis on their variance and higher moments.

Original language | English (US) |
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Pages (from-to) | 569-589 |

Number of pages | 21 |

Journal | Theory of Probability and its Applications |

Volume | 61 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- B-free integers
- Central limit theorem
- Correlation functions
- Entropy
- Möbius function

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## Cite this

Avdeeva, M., Cellarosi, F., & Sinai, Y. G. (2017). Ergodic and statistical properties of b-free numbers.

*Theory of Probability and its Applications*,*61*(4), 569-589. https://doi.org/10.1137/S0040585X97T988423