On the variance of squarefree integers in short intervals and arithmetic progressions

Ofir Gorodetsky; Kaisa Matomäki; Maksym Radziwiłł; Brad Rodgers

doi:10.1007/s00039-021-00557-5

On the variance of squarefree integers in short intervals and arithmetic progressions

Ofir Gorodetsky^*, Kaisa Matomäki, Maksym Radziwiłł, Brad Rodgers

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H< x⁶^/¹¹^-^ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q> x⁵^/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< x²^/³^-^ε and q> x¹^/³⁺^ε. Furthermore we show that obtaining a bound sharp up to factors of H^ε in the full range H< x¹^-^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 (US)
Pages (from-to)	111-149
Number of pages	39
Journal	Geometric and Functional Analysis
Volume	31
Issue number	1
DOIs	https://doi.org/10.1007/s00039-021-00557-5
State	Published - Feb 2021

Funding

We would like to thank Bingrong Huang and Francesco Cellarosi for useful conversations, and the anonymous referees for their helpful comments. OG was supported by the European Research Council (ERC) under the European Union\u2019s 2020 research and innovation programme (ERC Grant Agreement No. 786758). KM was supported by Academy of Finland Grant No. 285894. MR acknowledges partial support of a Sloan fellowship and of NSF Grant DMS-1902063. BR received partial support from NSF Grant DMS-1854398 and an NSERC grant. Parts of this research were done during visits to Centre de Recherches Math\u00E9matiques and Oberwolfach and we thank these institutions for their hospitality. We would like to thank Bingrong Huang and Francesco Cellarosi for useful conversations, and the anonymous referees for their helpful comments. OG was supported by the European Research Council (ERC) under the European Union?s 2020 research and innovation programme (ERC Grant Agreement No. 786758). KM was supported by Academy of Finland Grant No. 285894. MR acknowledges partial support of a Sloan fellowship and of NSF Grant DMS-1902063. BR received partial support from NSF Grant DMS-1854398 and an NSERC grant. Parts of this research were done during visits to Centre de Recherches Math?matiques and Oberwolfach and we thank these institutions for their hospitality.

ASJC Scopus subject areas

Analysis
Geometry and Topology

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Cite this

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title = "On the variance of squarefree integers in short intervals and arithmetic progressions",

abstract = "We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H< x6/11-ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q> x5/11+ε. On the assumption of respectively the Lindel{\"o}f Hypothesis and the Generalized Lindel{\"o}f Hypothesis we show that these ranges can be improved to respectively H< x2/3-ε and q> x1/3+ε. Furthermore we show that obtaining a bound sharp up to factors of Hε in the full range H< x1-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.",

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On the variance of squarefree integers in short intervals and arithmetic progressions

Abstract

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