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

Gorodetsky, Ofir and Matomäki, Kaisa and Radziwiłł, Maksym and Rodgers, Brad (2021) On the variance of squarefree integers in short intervals and arithmetic progressions. Geometric and Functional Analysis, 31 (1). pp. 111-149. ISSN 1016-443X. doi:10.1007/s00039-021-00557-5.

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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.

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Additional Information:© 2021 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 26 June 2020; Revised 18 November 2020; Accepted 09 December 2020; Published 31 March 2021. 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 Mathématiques and Oberwolfach and we thank these institutions for their hospitality.
Funding AgencyGrant Number
European Research Council (ERC)786758
Academy of Finland285894
Alfred P. Sloan FoundationUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Issue or Number:1
Record Number:CaltechAUTHORS:20210405-105931980
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Official Citation:Gorodetsky, O., Matomäki, K., Radziwiłł, M. et al. On the variance of squarefree integers in short intervals and arithmetic progressions. Geom. Funct. Anal. 31, 111–149 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108620
Deposited By: Tony Diaz
Deposited On:08 Apr 2021 22:50
Last Modified:21 Apr 2021 17:10

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