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Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Mangerel, Alexander P.

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Let ε>0 be sufficiently small and let 0<η<1/522 . We show that if X is large enough in terms of ε , then for any squarefree integer q≤X196/261−ε that is Xη -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression a(modq) , with (a,q)=1 . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which 196/261=0.75096… was replaced by 25/36=0.69 ¯ 4 . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the X3/4 -barrier for a density 1 set of Xη -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.


Mangerel, A. P. (2021). Squarefree Integers in Arithmetic Progressions to Smooth Moduli. Forum of Mathematics, Sigma, 9, Article e72.

Journal Article Type Article
Acceptance Date Oct 2, 2021
Online Publication Date Oct 27, 2021
Publication Date 2021
Deposit Date Oct 20, 2021
Publicly Available Date Jan 26, 2022
Journal Forum of Mathematics, Sigma
Print ISSN 2050-5094
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 9
Article Number e72
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Published Journal Article (797 Kb)

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Copyright Statement
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.<br /> <br /> &copy; The Author(s), 2021. Published by Cambridge University Press

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