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Multiplicative functions that are close to their mean

Klurman, Oleksiy and Mangerel, Alexander P. and Pohoata, Cosmin and Teräväinen, Joni (2021) Multiplicative functions that are close to their mean. Transactions of the American Mathematical Society, 374 (11). pp. 7967-7990. ISSN 0002-9947. doi:10.1090/tran/8427. https://resolver.caltech.edu/CaltechAUTHORS:20200110-150317815

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Abstract

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a multiplicative function f:N → {−1,1},lim sup x → ∞|∑n ≤ x μ²(n)f(n)|= ∞. This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplicative function f:N → C satisfies ∑n ≤ x f(n) = cx+O(1) with c ≠ 0 if and only if f(p) = 1 for all but finitely many primes and |f(p)|< 1 for the remaining primes. This answers a question of Ruzsa. For the case c = 0, we show, under the additional hypothesis ∑p 1−|f(p)|p < ∞,that f has bounded partial sums if and only if f(p) = χ(p)p^(it) for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f(p)| < 1. This provides progress on another problem of Ruzsa and gives a new and simpler proof of a stronger form of Chudakov’s conjecture. Along the way we obtain quantitative bounds for the discrepancy of the modified characters improving on the previous work of Borwein, Choi and Coons.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/tran/8427DOIArticle
https://www.ams.org/journals/tran/2021-374-11/S0002-9947-2021-08427-5/PublisherArticle
https://arxiv.org/abs/1911.06265arXivDiscussion Paper
ORCID:
AuthorORCID
Mangerel, Alexander P.0000-0003-4055-4078
Pohoata, Cosmin0000-0002-3757-2526
Teräväinen, Joni0000-0001-6258-8004
Additional Information:© 2021 American Mathematical Society. Received by the editors December 1, 2020, and, in revised form, February 18, 2021. Article electronically published on August 18, 2021. The fourth author was supported by a Titchmarsh Fellowship from the University of Oxford. The authors would like to warmly thank Marco Aymone for suggesting that our initial argument could lead to a proof of Theorem 1.1. We are grateful to the referee for a careful reading of the paper and many helpful comments. Finally, the first author greatly acknowledges support and excellent working conditions at the Max Planck Institute for Mathematics (Bonn).
Funders:
Funding AgencyGrant Number
University of OxfordUNSPECIFIED
Issue or Number:11
Classification Code:2020 Mathematics Subject Classification: Primary 11N37, 11N64
DOI:10.1090/tran/8427
Record Number:CaltechAUTHORS:20200110-150317815
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200110-150317815
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100641
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Jan 2020 00:51
Last Modified:17 Nov 2021 23:14

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