The Library
Moments of random multiplicative functions, II : high moments
Tools
Tools
Tools
Harper, Adam J. (2019) Moments of random multiplicative functions, II : high moments. Algebra and Number Theory, 13 (10). pp. 2277-2321. doi:10.2140/ant.2019.13.2277 ISSN 1937-0652.
|
PDF
WRAP-moments-random-multiplicative-functions-II-high moments-Harper-2019.pdf - Published Version - Requires a PDF viewer. Download (2224Kb) | Preview |
|
![[img]](https://wrap.warwick.ac.uk/style/images/fileicons/application_pdf.png) |
PDF
RMF high moments revision..pdf - Accepted Version Embargoed item. Restricted access to Repository staff only - Requires a PDF viewer. Download (567Kb) |
Official URL: http://dx.doi.org/10.2140/ant.2019.13.2277
Abstract
We determine the order of magnitude of $\E|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$.
In the Steinhaus case, we show that $\E|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by ``orthogonal'' rather than ``unitary'' behaviour. We also deduce some consequences for the large deviations of $\sum_{n \leq x} f(n)$.
The proofs use various tools, including hypercontractive inequalities, to connect $\E|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyse this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob's $L^p$ maximal inequality for martingales.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Journal or Publication Title: | Algebra and Number Theory | ||||||
| Publisher: | Mathematical Sciences Publishers | ||||||
| ISSN: | 1937-0652 | ||||||
| Official Date: | 6 January 2019 | ||||||
| Dates: |
|
||||||
| Volume: | 13 | ||||||
| Number: | 10 | ||||||
| Page Range: | pp. 2277-2321 | ||||||
| DOI: | 10.2140/ant.2019.13.2277 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | First published in Algebra & Number Theory in Vol. 13 (2019), No. 10, published by Mathematical Sciences Publishers © 2019 Mathematical Sciences Publishers | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 19 July 2019 | ||||||
| Date of first compliant Open Access: | 13 March 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
