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Almost sure large fluctuations of random multiplicative functions
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Harper, Adam J. (2023) Almost sure large fluctuations of random multiplicative functions. International Mathematics Research Notices, 2023 (3). pp. 2095-2138. doi:10.1093/imrn/rnab299 ISSN 1073-7928.
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WRAP-Almost-sure-large-fluctuations-random-multiplicative-functions-2021.pdf - Accepted Version - Requires a PDF viewer. Download (750Kb) | Preview |
Official URL: http://doi.org/10.1093/imrn/rnab299
Abstract
We prove that if f(n) is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of x for which |∑n≤xf(n)|≥x−−√(loglogx)1/4+o(1). This is the first such bound that grows faster than x−−√, answering a question of Halász and proving a conjecture of Erd̋s. It is plausible that the exponent 1/4 is sharp in this problem. The proofs work by establishing a multivariate Gaussian approximation for the sums ∑n≤xf(n) at a sequence of x, conditional on the behaviour of f(p) for all except the largest primes p. The most difficult aspect is showing that the conditional covariances of the sums are usually small, so the corresponding Gaussians are usually roughly independent. These covariances are related to a Euler product (or multiplicative chaos) type integral twisted by additive characters, which we study using various tools including mean value estimates for Dirichlet polynomials, high mixed moment estimates for random Euler products, and barrier arguments with random walks.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Multiplicity (Mathematics), Gaussian processes, Random variables | ||||||||
Journal or Publication Title: | International Mathematics Research Notices | ||||||||
Publisher: | Oxford University Press | ||||||||
ISSN: | 1073-7928 | ||||||||
Official Date: | February 2023 | ||||||||
Dates: |
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Volume: | 2023 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 2095-2138 | ||||||||
DOI: | 10.1093/imrn/rnab299 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Reuse Statement (publisher, data, author rights): | This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Adam J Harper, Almost Sure Large Fluctuations of Random Multiplicative Functions, International Mathematics Research Notices, 2021;, rnab299 is available online at: http://dx.doi.org/10.1093/imrn/rnab299 | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 1 December 2021 | ||||||||
Date of first compliant Open Access: | 2 November 2022 |
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