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Journal Article

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

MPS-Authors
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Heap,  Winston
Max Planck Institute for Mathematics, Max Planck Society;

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Zhao,  Jing
Max Planck Institute for Mathematics, Max Planck Society;

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Fulltext (public)

arXiv:2006.02754.pdf
(Preprint), 330KB

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There is no public supplementary material available
Citation

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.


Cite as: http://hdl.handle.net/21.11116/0000-0007-A24F-A
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.