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  Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

Aymone, M., Heap, W., & Zhao, J. (in press). Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function. Journal of the London Mathematical Society, Published Online - Print pending. doi:10.1112/jlms.12421.

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Item Permalink: http://hdl.handle.net/21.11116/0000-0007-A24F-A Version Permalink: http://hdl.handle.net/21.11116/0000-0007-A250-7
Genre: Journal Article

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arXiv:2006.02754.pdf (Preprint), 330KB
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https://doi.org/10.1112/jlms.12421 (Publisher version)
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 Creators:
Aymone, Marco, Author
Heap, Winston1, Author              
Zhao, Jing1, Author              
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory, Probability
 Abstract: We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Accepted / In Press
 Pages: -
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 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2006.02754
DOI: 10.1112/jlms.12421
 Degree: -

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Title: Journal of the London Mathematical Society
  Abbreviation : J. London Math. Soc.
Source Genre: Journal
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Publ. Info: Wiley
Pages: - Volume / Issue: - Sequence Number: Published Online - Print pending Start / End Page: - Identifier: -