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A new proof of Halász’s theorem, and its consequences

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Granville, Andrew, Harper, Adam and Soundararajan, K. (2019) A new proof of Halász’s theorem, and its consequences. Compositio Mathematica, 155 (1). pp. 126-163. doi:10.1112/S0010437X18007522

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Official URL: https://doi.org/10.1112/S0010437X18007522

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Abstract

Halász’s Theorem gives an upper bound for the mean value of a multiplicative function f. The bound is sharp for general such f, and, in particular, it implies that a multiplicative function with | f ( n ) |≤ 1 has either mean value 0, or is “close to” n it for some fixed t . The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s Theorem).

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Series, Arithmetic, Dirichlet series
Journal or Publication Title: Compositio Mathematica
Publisher: Cambridge University Press
ISSN: 0010-437X
Official Date: January 2019
Dates:
DateEvent
January 2019Published
23 November 2018Available
16 May 2018Accepted
Volume: 155
Number: 1
Page Range: pp. 126-163
DOI: 10.1112/S0010437X18007522
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: This article has been published in a revised form in Compositio Mathematica https://doi.org/10.1112/S0010437X18007522. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © The Authors 2018
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: The Author(s)
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
670239European Research Councilhttp://viaf.org/viaf/130022607
CRC program[NSERC] Natural Sciences and Engineering Research Council of Canadahttp://dx.doi.org/10.13039/501100000038
UNSPECIFIEDCentre de Recherches MathématiquesUNSPECIFIED
UNSPECIFIEDJesus College, University of Cambridgehttp://dx.doi.org/10.13039/501100000644
DMS 1500237National Science Foundationhttp://dx.doi.org/10.13039/100000001
UNSPECIFIEDSimons Foundationhttp://dx.doi.org/10.13039/100000893
DMS 1440140National Science Foundationhttp://dx.doi.org/10.13039/100000001
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