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Moments of random multiplicative functions, II : high moments
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Harper, Adam J. (2019) Moments of random multiplicative functions, II : high moments. Algebra and Number Theory .
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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 $\E\sum_{n \leq x} f(n)^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q1)^2}$ 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 > Mathematics  
Journal or Publication Title:  Algebra and Number Theory  
Publisher:  Mathematical Sciences Publishers  
ISSN:  19370652  
Official Date:  2019  
Dates: 


Date of first compliant deposit:  19 July 2019  
Status:  Peer Reviewed  
Publication Status:  Forthcoming  
Access rights to Published version:  Restricted or Subscription Access  
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