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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
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 | ||||||||||||||||||||||||
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| 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: |
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| 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) | ||||||||||||||||||||||||
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