## Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H< x^{6}^{/}^{11}^{-}^{ε} and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q> x^{5}^{/}^{11}^{+}^{ε}. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively H< x^{2}^{/}^{3}^{-}^{ε} and q> x^{1}^{/}^{3}^{+}^{ε}. Furthermore we show that obtaining a bound sharp up to factors of H^{ε} in the full range H< x^{1}^{-}^{ε} is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

Original language | English |
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Pages (from-to) | 111-149 |

Number of pages | 39 |

Journal | Geometric and Functional Analysis |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2021 |

Externally published | Yes |