The Library
Moments of random multiplicative functions, I : low moments, better than squareroot cancellation, and critical multiplicative chaos
Tools
Tools
Tools
Harper, Adam J. (2020) Moments of random multiplicative functions, I : low moments, better than squareroot cancellation, and critical multiplicative chaos. Forum of Mathematics, Pi, 8 . e1. doi:10.1017/fmp.2019.7 ISSN 2050-5086.
|
PDF
WRAP-moments-functions-squareroot-cancellation-chaos-Harper-2020.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (886Kb) | Preview |
|
![[img]](https://wrap.warwick.ac.uk/style/images/fileicons/application_pdf.png) |
PDF
WRAP-moments-random-multiplicative-functions-I-low-moments-better-squareroot-cancellation-critical-multiplicative-chaos-Harper-2019.pdf - Accepted Version Embargoed item. Restricted access to Repository staff only - Requires a PDF viewer. Download (1299Kb) |
Official URL: https://doi.org/10.1017/fmp.2019.7
Abstract
We determine the order of magnitude of E|∑n≤xf(n)|2q, where f(n) is a Steinhaus or Rademacher random multiplicative function, and 0≤q≤1. In the Steinhaus case, this is equivalent to determining the order of limT→∞1T∫T0|∑n≤xn−it|2qdt. In particular, we find that E|∑n≤xf(n)|≍x−−√/(loglogx)1/4. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment, and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of ∑n≤xf(n). The proofs develop a connection between E|∑n≤xf(n)|2q and the q-th moment of a critical, approximately Gaussian, multiplicative chaos, and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Journal or Publication Title: | Forum of Mathematics, Pi | ||||||
| Publisher: | Cambridge University Press | ||||||
| ISSN: | 2050-5086 | ||||||
| Official Date: | 20 January 2020 | ||||||
| Dates: |
|
||||||
| Volume: | 8 | ||||||
| Article Number: | e1 | ||||||
| DOI: | 10.1017/fmp.2019.7 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | This article has been published in a revised form in Forum of Mathematics, Pi [ https://doi.org/10.1017/fmp.2019.7 ]. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge University Press 2019. | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Copyright Holders: | The Author | ||||||
| Date of first compliant deposit: | 13 November 2019 | ||||||
| Date of first compliant Open Access: | 3 February 2020 | ||||||
| Related URLs: |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
