A more intuitive proof of a sharp version of Halász’s theorem
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- by Andrew Granville, Adam J. Harper and Kannan Soundararajan PDF
- Proc. Amer. Math. Soc. 146 (2018), 4099-4104 Request permission
Abstract:
We prove a sharp version of Halász’s theorem on sums $\sum _{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the “average of averages” and “integration over $\alpha$” manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method, we express $\sum _{n \leq x} f(n)$ as a triple Dirichlet convolution and apply Perron’s formula.References
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Additional Information
- Andrew Granville
- Affiliation: Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada – and – Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England
- MR Author ID: 76180
- ORCID: 0000-0001-8088-1247
- Email: andrew@dms.umontreal.ca
- Adam J. Harper
- Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, England
- MR Author ID: 871455
- Email: A.Harper@warwick.ac.uk
- Kannan Soundararajan
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 319775
- Email: ksound@stanford.edu
- Received by editor(s): July 14, 2017
- Received by editor(s) in revised form: December 27, 2017
- Published electronically: May 15, 2018
- Additional Notes: The first author received funding in aid of this research from the European Research Council grant agreement no. 670239 and from NSERC Canada under the CRC program.
The second author was supported, for parts of the research, by a postdoctoral fellowship from the Centre de recherches mathématiques in Montréal and by a research fellowship at Jesus College, Cambridge.
The third author was partially supported by NSF grant DMS 1500237 and a Simons Investigator grant from the Simons Foundation. - Communicated by: Matthew A. Papanikolas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4099-4104
- MSC (2010): Primary 11N37, 11N56, 11N64
- DOI: https://doi.org/10.1090/proc/14095
- MathSciNet review: 3834641